System tool for weight optimization of thin wall structures

ABSTRACT

A thin wall profile member (TPM) with a cross-sectional shape and dimensions constant along its length is pre-constructed for use as a load carrying, weight optimized structural element. The TPM has at least one main strip having longitudinal reinforcing ribs and at least one additional strip having longitudinal reinforcing ribs. The at least one additional strip has a width that does not exceed a width of the at least one main strip and has a thickness that equals or exceeds a thickness of the at least one main strip. A ratio of the width of the at least one additional strip to the width of the at least one main strip is in the range of 0.05 to 1.0. A ratio of the thickness of the at least one additional strip to the thickness of the at least one main strip is in the range of 1.0 to 5.0.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related, but does not claim benefit of priorityfrom, continuation-in-part (CIP) of patent application Ser. No.12/462,521, filed 5 Aug. 2009, which is a continuation of patentapplication Ser. No. 10/913,616, filed 6 Aug. 2004, which is acontinuation of patent application Ser. No. 10/149,049, filed 4 Jun.2002, which is the National Stage of International Application No.PCT/RU 00/00494, filed 1 Dec. 2000. It is further related, but does notclaim priority from, CIP application Ser. No. 13/841,700 and CIPapplication Ser. No. 13/842,488, filed even date herewith, and whichclaim priority from one or more of the aforementioned relatedapplications, all of which applications are commonly owned and assigned.

BACKGROUND OF THE INVENTION Field of the Invention

The present disclosure is related to modeling and analysis tools fordesign and selection of weight-optimized structural members subject toload-bearing forces.

Background Information

Compressed thin wall structural members (“thin wall structures”) arewell suited for a variety of applications—ranging from constructionprojects to mechanical equipment and machinery design—that require lowweight, high strength and high stability and flexibility.

The basic principles of thin wall structural design optimization thathave important practical significance for solving the inverse problem ofstructural mechanics are described in F. R. Shanley's textbook, WeightAnd Strength Analysis In Aircraft Design. In this work, problems of thinwall tubular column optimization and problems of designing a plate andtubular shell affected by the bending moment are solved; the issue ofoptimal installation of reinforcement plates (stringers and ribs) isaddressed. These examples describe approaches to solving multi-parameteroptimization problems based on use of fundamental principles of platesand bars structural theories applied to simple optimization techniques.Optimization techniques are selected based on analysis of performanceand stability failure modes of compressive structural members. Theobjective of Shanley's approach was to establish a functional connectionbetween the allowable stress value, on the one hand, and geometricparameters and the external load, on the other.

Because such a connection is rather difficult to establish due to themulti-parameter nature of the task, the number of variables needs to bereduced by expressing certain variables using an integrating factor. Aneffectively selected integrating factor can be a criterion of theminimal cross-section area (minimum weight) if actual stability failuremodes of the compressive shape are considered. In this case, the maximumvalue of the integrating factor equals the minimum value of thecross-section area. If the thin wall shape buckles only in overallstability mode, then the integrating factor, named the “shape factor” byShanley (K_(F)=i²/F, where i and F are the radius of gyration and barcross section area, respectively), can serve as the minimum weightcriterion. If the compressive shape is also subject to local buckling,then the “shape factor” cannot serve as the criterion of minimum weightbecause its increase will result in reduction of the shape's bearingcapacity (due to decrease in the local buckling critical stress).

It is a great challenge, from a design standpoint, to achieve optimumweight thin wall structures, especially, when one must also account formaximum expected load-bearing capacity along several cross-sections anddifferent materials.

Thin wall structures are traditionally formed from thin wall profilemembers (TPMs) generally referred to as thin wall shapes. There are manyapplications where thin wall structures, due to low weight, are the onlydesign choice for engineers and designers.

In some instances, in order to enhance the load-bearing capacity of thinwall shapes, the same TPMs may be “reinforced’ using reinforcing platesand/or shells (collectively referred to as “panels”). For purposes ofdiscussion, we shall refer to reinforced TPMs as “TPM-panelcombinations” or simply “TPM-panels”.

Structural design optimization has been developed, in general, for suchsimple structural members as thick wall beams, heavy cross-section bars,trusses, arches and frames. Thin wall plates, shells and shapes (TPMs)are significantly less researched.

In the same way as designing thick structures, thin wall structures mustbe designed to satisfy a variety of “constructive restrictions.” Forexample, the end design product must “fit” or “accommodate” the physicalspace for which it is designed. Other constructive restrictions mayinclude, depending on the application, additional criteria such asweight, strength, stability, and flexibility constraints, as well asfactors having to do with temperature (heat/cold resistance),conductivity, and other well-known material-selection criteria.

Modern design optimization relies heavily on computers which model andanalyze a configuration to ensure that a particular design is suitable.Conventional tools, however, rely heavily on modeling techniques thatdraw mainly from thick wall structures, and which may be used forexample to select an I-beam for a bridge or other extremely heavy loadbearing project, where weight optimization, while important, it is notas critical as applications involving for example, aircrafts components.

Consequently, modern design of thin wall structures is focused more onsingle parameter-by-parameter optimization, with little emphasis oninter-dependencies of these parameters to other parameters. For example,critical stress, external compressed load, material properties,cross-section dimensions, and other criteria all impact, separately andinterdependently, optimum design configuration and material selection.

Because inter-dependencies are not properly mapped out with the goal ofweight optimizing a design for a given constructive restriction, theprocess is relatively unsophisticated.

It is desirable to be able to improve weight-optimization processes indesign particularly in applications where TPMs may be employed.

SUMMARY

The present disclosure describes tools and associated computationalanalysis methodologies employed therein for improved TMP and TMP-panelweight-optimization and selection.

The tools draw on inter-dependence parameters relating to TPMcross-section dimensions ratio values and established constructiverestrictions to calculate—using appropriate algorithmic computationalanalysis—the optimum cross-section dimensions values of a given TPM. Theproposed computational analysis is used in the selection ofweight-optimized TPM-panels.

A design selection serves as a blueprint for the next stage which is theactual fabrication or manufacturing of the component. For a given set ofconstructive restrictions, the final product is based on optimumconfigurations selected from a fixed set of TMPs with variedcross-section shapes; or in the case of TPM-panels, on optimumconfigurations selected from a fixed set of TMP-panel combinations ofvaried cross-section dimensions.

In a further exemplary embodiment, a proposed methodology and associatedsystem tool are described that decrease the number of variables inoptimization of a wing box, the load carrying components of which areTPM shapes and TPM-panels.

BRIEF DESCRIPTION OF THE DRAWINGS

The appended drawings illustrate exemplary configurations of thedisclosure and, as such, should not be considered as limiting the scopeof the disclosure that may admit to other equally effectiveconfigurations. Correspondingly, it has been contemplated that featuresof some configurations may be beneficially incorporated in otherconfigurations without further recitation.

FIGS. 1-8 depict TPMs of various configurations, each characterized by amain strip and having a dimensional component (width b), in accordancewith an exemplary embodiment.

FIG. 9 diagrammatically maps shape efficiency factor value E as afunction of width b.

FIG. 10 is a cross-section perspective of a TMP-panel combination inaccordance with a further embodiment of the present invention.

FIG. 11 is an inverted perspective of the configuration in FIG. 10showing the I- and L-shaped TPMs positioning on the panel which functionto simultaneously reinforce the panel in the longitudinal and transversedirections, respectively.

FIG. 12 is a sectional view of FIG. 11 taken along line “A-A”.

FIG. 13 is a further sectional view as seen in the direction of arrow“B” in FIG. 11.

FIG. 14 is a cross-sectional perspective of a further TPM-panelconfiguration defined by a U-shaped TPM in accordance with a furtherembodiment;

FIG. 15 is a further cross-sectional perspective of a TPM-panelconfiguration substantially as shown in FIG. 14 but including, inaddition to the U-shaped TPM, a further Z-shaped TPM to cooperativelyreinforce the panel in both longitudinal and transverse directions,respectively.

FIG. 16 is a flow diagram describing the operational flow of a model andanalysis TPM design tool configured for selecting an optimum weight TPMin accordance with an exemplary embodiment.

FIG. 17 is a flow diagram describing the operational flow of a model andanalysis TPM design tool configured for selecting an optimum weightTPM-panel in accordance with a further exemplary embodiment.

FIG. 18 is a plot showing factor Ks versus ratios of U-shaped TPMdimensions.

FIG. 19 is a another plot showing factor Ks versus ratios of U-shapedTPM dimensions.

FIG. 20 is a yet another plot showing factor Ks versus ratios ofU-shaped TPM dimensions.

FIG. 21 is a plot showing factor Ks versus ratios of Z-shaped TPMdimensions.

FIG. 22 is a plot showing factor Ks versus ratios of L-shaped TPMdimensions.

FIG. 23 is a plot showing factor Ks versus ratios of I-shaped TPMdimensions.

FIG. 24 is a plot of shape efficiency factor A versus ratios of U-shapedTPM dimensions.

FIG. 25 is another plot showing shape efficiency factor A versus ratiosof U-shaped TPM dimensions.

FIG. 26 is yet another plot showing shape efficiency factor A versusratios of U-shaped TPM dimensions.

FIG. 27 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{a} = \frac{a}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

FIG. 28 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{\delta_{a}} = \frac{\delta_{a}}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

FIG. 29 is a plot showing optimum stress σ_(s) versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

FIG. 30 is a plot showing dimensionless weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

FIG. 31 is a plot showing shape efficiency factor A versus ratios ofZ-shaped TPM dimensions.

FIG. 32 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{b} = \frac{b}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 33 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{\delta_{b}} = \frac{\delta_{b}}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 34 is a plot showing optimum stress σ_(s) versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 35 is a plot showing dimensionless weight versus

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 36 is a plot showing shape efficiency factor A versus ratio ofL-shaped TPM dimensions.

FIG. 37 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{b} = \frac{b}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

FIG. 38 is a plot showing scaled (dimensionless) characteristicdimension

${\overset{\_}{\delta}}_{b} = \frac{\delta_{b}}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

FIG. 39 is a plot showing optimum stress σ_(s) versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

FIG. 40 is a plot showing dimensionless weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

FIG. 41 is a table showing feasible dimensions of U-shaped TPM.

FIG. 42 shows a panel with Z-shaped TPM.

FIG. 43 is a plot showing dimensionless panel weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus TPM stress σ_(s), U-shaped TPM.

FIG. 44 is a plot showing dimensionless skin thickness

$\overset{\_}{\delta} = \frac{\delta}{l_{re}}$

versus the panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

FIG. 45 is a stringer panel with U-shaped TPM.

FIG. 46 is a plot showing dimensionless distance between stringers

${\overset{\_}{b}}_{s} = \frac{b_{s}}{l_{re}}$

versus the panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

FIG. 47 is a plot showing TPM shape width a, b, c versus the panelstress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

FIG. 48 is a plot showing TPM shape thickness δ_(a), δ_(b), δ_(c) versusthe panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

FIG. 49 is a plot showing dimensionless panel weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus the panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

materials D16-T and B95 (least feasible weight panel).

FIG. 50 is a plot showing dimensionless distance between stringers

${\overset{\_}{b}}_{s} = \frac{b_{s}}{l_{re}}$

versus the panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (least feasible weight panel).

FIG. 51 is a plot showing dimensionless length of TPM shapes

${\overset{\_}{a} = \frac{a}{l_{re}}},$

$\overset{\_}{b} = \frac{b}{l_{re}}$

versus the panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (least feasible weight panel).

FIG. 52 is a plot showing dimensionless length of TPM shapes

${\overset{\_}{b} = \frac{b}{l_{re}}},{\overset{\_}{c} = \frac{c}{l_{re}}}$

versus the panel stress facto

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (least feasible weight panel).

FIG. 53 is a plot showing dimensionless thickness of U-shaped TPM

${\overset{\_}{\delta_{a}} = \frac{\delta_{a}}{l_{re}}},{\overset{\_}{\delta_{b}} = \frac{\delta_{b}}{l_{re}}},{\overset{\_}{\delta_{c}} = \frac{{\overset{\_}{\delta}}_{c}}{l_{re}}}$

versus panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}^{2}}},$

material D16-T (least feasible weight panel).

FIG. 54 is a plot showing dimensionless thickness of TPM

${\overset{\_}{\delta_{b}} = \frac{\delta_{b}}{l_{re}}},{\overset{\_}{\delta_{c}} = \frac{\delta_{c}}{l_{re}}}$

versus panel stress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}^{2}}},$

material D16-T (least feasible weight panel).

FIG. 55 is a stringer panel with U-shaped TPM possessing inclinedlateral web.

FIG. 56 shows real and optimum-design stringer panels.

FIG. 57 are tables showing (i) a range of allowable values for stringerpanel stress factors, and (ii) features of real and optimum-designpanels.

FIG. 58 shows a rib-reinforced stringer panel.

FIG. 59 shows scheme for determining of the critical rigidity of ribs.

FIG. 60 shows scheme for determining of the rigidity of ribs.

FIG. 61 is a plot of dimensionless weight of the stringer panel

${{\overset{\_}{G}}_{p \cdot s} = \frac{G_{p \cdot s}}{l \cdot \gamma}},$

ribs

${\overset{\_}{G}}_{r} = \frac{G_{r}}{l \cdot \gamma}$

and rib-reinforced stringer panel G _(p.r)=G _(p.s)+G _(r) versus panelstress σ, material D16-T.

FIG. 62 is a plot of dimensionless weight of the rib-reinforced stringerpanel

${\overset{\_}{G}}_{p \cdot s} = \frac{G_{p \cdot s}}{l \cdot \gamma}$

versus distance between ribs l_(r), material D16-T.

FIG. 63 is a plot of dimensionless weight of the rib-reinforced stringerpanel

${\overset{\_}{G}}_{p \cdot r} = \frac{G_{p \cdot r}}{l \cdot \gamma}$

versus skin thickness δ, material D16-T.

FIG. 64 is a plot of dimensionless weight

${\overset{\_}{G}}_{p \cdot r} = \frac{G_{p \cdot r}}{l \cdot \gamma}$

of the rib-reinforced stringer panel versus panel stress σ with varyingrib rigidity factor.

FIG. 65 is a plot of dimensionless weight

${\overset{\_}{G}}_{p.r} = \frac{G_{p.r}}{l \cdot \gamma}$

of the rib-reinforced stringer panel versus panel stress σ with varyingrib shape factor f_(r), material D16-T.

FIG. 66 plots skin thickness δ of the rib-reinforced stringer panelversus linear compressive force q_(p).

FIG. 67 plots distance between stringers b_(s) of the rib-reinforcedstringer panel versus linear compressive force q_(p).

FIG. 68 plots distance between ribs l_(r) of the rib-reinforced stringerpanel versus linear compressive force q_(p).

FIG. 69 plots characteristic dimensions of stringer shape a, b of therib-reinforced stringer panel versus linear compressive force q_(p).

FIG. 70 plots characteristic thicknesses of stringer shape δ_(a), δ_(b)of the rib-reinforced stringer panel versus linear compressive forceq_(p).

FIG. 71 plots optimal stress σ of the rib-reinforced stringer panelversus linear compressive force q_(p).

FIG. 72 plots dimensionless weight

${\overset{\_}{G}}_{p.r} = \frac{G_{p.r}}{l \cdot \gamma}$

of the rib-reinforced stringer panel versus linear compressive forceq_(p).

FIG. 73 plots aerodynamic profile of the wing with the specifiedposition of the torsion box.

FIG. 74 shows a wing design scheme.

FIG. 75 is a table showing coefficients K_(σ.w), K_(τ.w) in the formulaefor web cell critical stresses.

FIG. 76 is a table showing factor θ versus web cell overall dimensions.

FIG. 77 is a table showing feasible coefficients for a stringer shape.

FIG. 78 is a table showing dimensions and geometrical characteristics ofrib shape.

FIG. 79 is a table showing dimensions and geometrical characteristics ofpillar shape.

DETAILED DESCRIPTION

The detailed description set forth below in connection with the appendeddrawings is intended as a description of exemplary embodiments of thepresent invention and is not intended to represent the only embodimentsin which the present invention can be practiced. The term “exemplary”used throughout this description means “serving as an example, instance,or illustration,” and should not necessarily be construed as preferredor advantageous over other exemplary embodiments. The detaileddescription includes specific details for the purpose of providing athorough understanding of the exemplary embodiments of the invention. Itwill be apparent to those skilled in the art that the exemplaryembodiments of the invention may be practiced without these specificdetails. In some instances, well known structures and devices are shownin schematic form with no additional detail in order to avoid obscuringthe novelty of the exemplary embodiments presented herein.

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments.

The subject matter of the present inventions may best be understood byreference to the following descriptions taken in connection with theaccompanying drawings.

The present disclosure describes improved TMP and TMP-panel designoptimization techniques. The techniques utilize advanced mathematicalmodeling techniques to perform practical multi-parameter associations.

The term “constructive restrictions” is used herein to refer to TPMdimensionality necessary to achieve desired structural, material andfabrication requirements and specifications.

Structural requirements for load-carrying TPMs and TPM-panelcombinations are set to ensure stability and stiffness under compressiveload. An excessive compression load could, for example, lead tobuckling, an unstable condition affecting the TPM but also potentiallythe surrounding environment.

Material requirements for load-carrying TPMs and TPM-panel combinationsare set to ensure minimum weight objectives at maximum expectedload-bearing capacity.

Fabrication requirements for load-carrying TPMs and TPM-panelcombinations are typically associated with manufacture or designspecific parameters relating to sizing, cost, proper fit, etc. Anexample set of fabrication type requirements may be requirements thatcall for minimum thickness of TPM to allow mass production, providing apanel sheet thickness which conforms to standard thickness values, andensuring that any design allows for any flanges incorporated thereon,for example, to allow the mounting of rivets.

For purposes of the present disclosure, a constructive restriction isany variable which plays any role whatsoever in the ability to achievethe best possible weight optimization limited only by the available ofTPM and/or TPM-panel configurations available for selection by adesigner as part of the design process.

Glossary of Terms to be Used

Bow Longitudinal curvatures of rod, bar, shapes and tube. Buckle Adistortion of the surface of the material. Elasticity The ability of amaterial or body to return to its original shape and dimensions afterbeing deformed by stress. Profile/Shape A product that is long inrelation to its cross- sectional dimensions (not a wire, a rod, or abar), formerly termed a “shape”. Proportionality The maximum stresswhich a metal still obeys limit stress Hooke's Law. Structural shape Asection of any design accepted as standard by the structural industry;such shape include I-beam, wide flange of H-beam, channels, angles, teesand zees. Ultimate stress The maximum stress (tensile, compressive, orshear) that a material can withstand. Yield stress The stress at which amaterial exhibits a speci- fied permanent set. For aluminum allow andsteel is 0.2% of gauge length; for aluminum allow the yield stresses intension and compression are approximately equal. Thin wall Threedimensions of these structures are expressed structures by three valuesof different magnitude (shapes, tubes, wires, rods, bars), or two out ofthree dimensions of such structures are expressed by values of the samemagnitude, and the third one, related to thickness, represents a smallervalue with respect to those two (thin plate, shells, thick planks). Thebasic ultimate condition of thin wall structures is buckling undercompressive load.

Nomenclature

Structural Property Notation M bending moment in calculated wingsection, [daN · cm] M_(ts) or M_(tor) torsion moment in calculated wingsection, [daN · cm] Q lateral or transversal force in calculated wingsection, [daN] q uniform or linear compressive force, [daN/cm] Pconcentrated compressive force, [daN] l distance between calculated wingsections, [cm] B width of box beam in calculated wing section, [cm]c_(max) maximum thickness of wing aerodynamic profile, [cm] h_(ef)effective height of wing box, [cm] h_(w) height of wing box sectionvertical wall, [cm] δ thickness of shape-stiffened plate, [cm] tthickness of vertical wall of wing box section, [cm] b_(s) longitudinalinstallation shape spacing (stringer spacing), [cm] b_(re) reduced widthof shape-stiffened plate, [cm] l_(s) shape length, [cm] l_(re) shapeeffective length, l_(re) = l_(s)/√c_(s,) [cm] l_(r) lateral installationshape spacing (rib spacing), [cm] r pillar spacing (pillars supportingvertical walls), [cm] a, δ_(a), b, shape cross-section dimensions, [cm]δ_(b), c, δ_(c) F shape cross-section area, [cm²] J shape moment ofinertia, [cm⁴] i_(s) shape radius of gyration, i_(s) = √(J/F), [cm] Wwing box moment of resistance [cm³] Ω wing box area, [cm²] σ, τ normaland shear stress, [daN/cm²] P _(c) shape load intensity or stressfactor, P _(c) = P/l_(re) ², [daN/cm²] q _(p) panel load intensity orstress factor, q _(p) = P/(B/l_(re)) = q/l_(re), [daN/cm²] MaterialProperty Notation σ_(pr), τ_(pr) compression and shear proportionalitylimit stress, [daN/cm²] σ_(0.2), τ_(0.2) compression and shear yield(0.2% offset) stress, [daN/cm²] σ_(b), τ_(b) tensile and shear ultimatestress, [daN/cm²] σ₀, τ₀ compression and shear limiting stress,[daN/cm²] E compression normal modulus (Young's modulus), [daN/cm²]G_(s.p.) or G_(p.s) stringer panel weight, [kg] G_(p.r.) or G_(r.p) ribstringer panel weight, [kg] G_(w) pillar wall weight, [kg] G wing boxbeam weight, [kg] γ density of material, [kg/cm³] Non DimensionalFactors f shape factor η_(σ), η_(τ) compression and shear plasticity orelastic ductility factors B rib strength factor K_(e) bending stiffnessrib factor A, Σ shape efficiency factor K_(σ), K_(τ) compression andshear crippling cell factors Θ pillar moment of inertia factor Ψ shapedimensions variation factor c_(s) stringer end fixity factor Sub - andsuperscripts ts torsion cr critical l local tot total (overall) r rib skskin (plate) p panel s stringer pi pillar w wall ef effective re reducedpr proportional s.p. or p.s. stringer panel r.p. or p.r. ribbed panel exexperimental op optimum n normative (feasible) rq required riv rivet avaverage tr transversal d design

INTRODUCTION

A structural design optimization methodology and tool is hereindescribed. The technology pertains to thin wall structures in respect ofminimum weight criterion. Technology emphasis is the weight optimizationof structural metal thin wall load bearing shapes of variousconfigurations and panels. Intended uses include applications spanningthe field from aerospace, construction, transportation and otherindustries.

The technology aims in part to take advantage of the widespreadcommercial availability of extruded (rolled) thin wall structures whichcome in a wide variety of standards and sizes, with H, L, Z, I, C, T,and U-shaped configurations being widely deployed for this reason.Aluminum alloy and steel structures are particularly plentiful andcommon design choices by designers across industries with load bearingdesign needs.

The aim of structural design optimization is to be able to achievereliable thin wall structures which are at the same time light in weightand lower cost if possible.

A software tool capable of addressing the inefficiencies and lack ofprecision in prevailing structural optimization processes would bequickly embraced by structural engineers, designers and developers.

The presently proposed solution solves inverse task of structuralmechanics for TPMs and TPM-panels: given load, material, pattern ofaxes, overall size (length, width), and constructive restrictions on thestructure, cross-section dimensions are found corresponding to theoptimum weight structure.

In a real world environment, a structural design optimization isconsidered solved when demonstrating changes of weight value in a widerange of structural parameters for different types of structures andmaterials. For instance, it is not sufficient only to find out that theoptimum height or thickness of the shape corresponds to their optimumvalues. Because the calculated optimum values may not be implemented dueto structural, material, or fabricated restrictions, it is equallyimportant to demonstrate how optimum weight values will change forothers shapes, in view of these restrictions. The designer should notonly select the best suited material, but he should also know how thestructure's weight will change in case a different material is used.Because the basic ultimate condition of thin wall structures is bucklingunder compressive load, these data cannot be obtained by directcomparison of mechanical properties specifications of materials, becausethe weight of buckling thin wall structure depends on the compressiveload value when the material's mechanical properties are changed.

Modern structural optimization design concepts rely heavily onmathematical modeling, using as principles, functional analysis theory,theory of random functions, and the theory of operations.

Many structural optimization concepts however suffer from an “academic”approach to addressing the problem. Traditional approaches rely onsingle-discipline optimization, at best coupling low accurate modelingof other relevant disciplines during the detailed design phase withoutconsideration of all design variables and restrictions. Omission or poorformulation of even one restriction leads to undesirable effects anddismissal of the result as irrelevant.

It would be very much desirable to be able to make generalization thaton the one hand make it possible to single out optimization methods thatcombine certain main conditions from perspective of strength, overalland local buckling, stiffness, according to the industry standards,while at the same time, leave room for structural, material, and/orfabricated restrictions in arriving at optimum design configurations andparameters.

Because the final output of a designer's work is a structural blueprint,design methodologies should allow being able to determine optimumsuitable configurations based in large extent on profile ofcross-sections and cross-sectional dimensions of structures.

Because of high labor intensity of computations, getting accurateresults of simultaneous optimization even of three- or four-parameterproblems is practically possible only by utilization of significantcomputer resources and time. Several search methods for the optimum areknown of multi-parameter problem solving using a computer: classicalanalysis method (variation calculation); deterministicapproach—scanning, mathematical programming (gradient based methods);probabilistic (random) search. The variation calculation method requiressolving an equations system with partial derivatives for variableparameters. The disadvantages of this method are: cumbersome equationsand—more importantly—this method do not allow any discontinuities orfunctions' extreme points and their derivatives. The latter does notallow using this method for optimization of some parameters. In thedeterministic approach, the gradient method is a two-step one: first,partial derivatives for variable parameters are calculated. Then, thesystem shifts in the direction opposite to the gradient one (the scalarfunction gradient vector is co-directional with this function's largestincrease). The gradient method is well suited to solving linearproblems. The majority of practical parameter optimization problems arenonlinear ones. In this case, the method is time consuming.

Structural design optimization has been developed in general for suchsimple structural members as thick wall beams, weighted cross-sectionbars, trusses, arches, and frames. Thin wall plates, shells, and shapesare significantly less researched. Only those inverse tasks ofstructural mechanics that have the simplest mathematical models areexplored in the thin wall structural design optimization.

Traditional approaches for determining optimum geometrical parameters isto use cross-section area as variable parameters rather thancross-section dimensions (perhaps allowing reduction of number ofvariables.) Local stability is ensured by restricting thecross-sectional dimensions based on the results of the separate localstability problem solved with the consideration of design issues inmind.

The constructive restrictions are taken into consideration whencross-section dimension values are calculated based on the results ofthe area optimization problem by using the iterative search technique.However, iterative search does not allow one to determine how close theselected set of cross-section dimensions is to the optimum one, neitherdoes it ensure correspondence to the minimum cross-sectional area.Therefore, such problem formulation is less accurate than whencross-section dimensions are assumed as variables during theoptimization process.

For example, Boeing Design Manual (BDM-6080) on optimizing round tubecolumns of different aluminum alloys at combined buckling and cripplingmodes does not adapt the disclosed approach to the more complexstructures BDM optimizes.

In one scenario, BDM notes “a Z section has no common relationship suchas D/t” and enforces “some geometric constraints to dictate its optimumdimensions,” declaring that “In the physical behavior of the Z section,however, this is difficult to achieve.” In BDM, local stability(crippling) is ensured by restricting the cross-sectional dimensionsratios based on the results of the separate local stability problemsolved with cross-section area as a variable.

The presently proposed approach, described in greater detail below, hasshown great promise in design optimization of practical multi-parameterthin wall structures using only mathematical analysis and modeling.

An important distinction in the proposed approach is in its reducing thedimensionality of a problem (by decreasing the number of simultaneouslyvariable parameters) by employing a general scheme and incorporatingcommon design approaches. This is achieved in principle by an initialprocess of:

1) transition from simple calculated schemes to more complex ones basedon the analysis of work and different stability failure modes in astructure; 2) introduction of integrating factors—design indexes; 3)introduction constructive restrictions; and 4) use of the equalstability principle.

After this initial process is completed, a scanning methodology isemployed (e.g., direct enumeration methodology) and a small number ofresulting variables used. The proposed process was found to make itpossible to scale down the optimization process of a wing box, forexample, from 20 to 4 variable geometric parameters. This is describedand explained further below.

The proposed solution does require a high level of familiarity andadvanced knowhow of load and material science. During aircraft design,the present methodology may be utilized during the advanced detaileddesign stage.

A better practice would be to identify optimum weight structure shapedesigns and configurations at the earliest possible stage indesign—which in the case of aircraft, is the preliminary and conceptdesign phase. This also leaves room for being able to make more informeddecisions on matters of design and, of course, to reduce wastefulefforts.

Using the extensive data from practical application of the technology todifferent shapes, materials, and constructive restrictions and analysisof plots of the new geometric index, ranges of cross-section dimensionratio values critical for optimum weight have been established. For eachTPM shape, inside the ranges, the values of the shape efficiency factorhave insignificant deviation from its theoretical maximal values andthus insignificant deviation from its theoretical minimum weight values.Therefore, constructive restrictions to actual cross-section dimensionsneed be factored into the ratio values inside the ranges to achieveoptimum weight.

Outside the ranges, the weight of the shapes increases significantly.The ranges are the same for all loads, materials, shapes, andconstructive restrictions and serve as an essential criterion of qualityand efficiency of the developed design.

By further reducing the parameters of the optimization task, the rangescan be used early in design to assure a minimum weight structure thatwill perform with integrity under any load and using any material. Theranges enable decreased labor intensity and simplify computationstandardization as well as development of standard optimum designprograms and design manuals.

Model and Analysis Tool—High Level Description

The present invention employs a software-based, model and analysis toolthat performs computational analysis on designer-selected TPMs to helpin the selection of the appropriate TPM with minimum weight and size fora particular application.

The analysis uses criteria that have never been used before to select aTPM of optimum weight, and in fact, for this reason, it is believed thatconventional techniques do not provide an opportunity for a designer tobe able to use computationally-determined solutions to ensure that aparticular TPM or TPM-panel is weight-optimized for a given applicationor use.

The proposed approach employs a process whereby the number ofsimultaneously relevant variables under consideration are decreasedusing a number of design approaches. One such design approach involves aprocess whereby, starting with an initial structure, a modeling andsimulation process is initiated which applies work and stability failureanalysis, starting with initial simple calculation methodologies andtransitioning to higher level of calculation complexity. In parallel orsubsequent thereto, (i) integrated factors-design indexes, (ii) equalstability principle, and (iii) constructive restriction parameters areall introduced.

The design output serves as a blueprint for the next stage which is theactual fabrication or manufacturing of the component. For a given set ofconstructive restrictions, the final product is based on optimumconfigurations selected from a fixed set of TMPs with variedcross-sectional shapes; or in the case of TPM-panels, on optimumconfigurations selected from a fixed set of TMP-panel combinations ofvaried cross-sectional dimensions.

The present disclosure uses mathematical modeling to weight-optimizedesign process without using conventional iterative process analysis andmodeling.

Where TPMs have been deployed with selected cross-section dimensions, atno time has the effect of “spacing” of cross section material been takeninto consideration in determining optimum weight. As a result, at ahigher moment of inertia, while higher overall stability is achieved,local stability is reduced.

TPM-panel combination arrangements are also known, but as has alreadybeen explained, design tools all rely on iterative modeling techniqueswhich suffer the drawbacks already described.

The present disclosure describes new and novel techniques for selectingoptimum design TPMs and TPM-panel combinations that meet constructiverestrictions that are optimized for weight.

The proposed approach is a system of optimization including a problemformulation strategy that is based on applying the inverse task ofstructural mechanics. The system minimizes the number of parametersvaried simultaneously, thereby reducing time to design as well as costof design while a weight optimized solution is achieved.

The proposed system tool is a promising solution for practicalmulti-parameter thin wall structure optimization problems completelycaptured mathematically. The system serves to reduce the dimensionalityof the problem (decrease the number of simultaneously variedparameters), thereby achieving savings in time and cost of design, byusing common design approaches such as:

1) transition from simple calculation schemes to more complex ones basedon the analysis of work and different stability failure modes in astructure, 2) introduction of integrating factors—design indexes; 3)introduction of constructive restrictions, and 4) use of the equalstability principle.

A method of scanning (direct enumeration method) a small number ofresulting variables is employed. This method of scanning is the singleglobal optimum search method (direct enumeration method). It does notrequire any special type of functions connecting the sought parameterswith the assessment criterion (optimum weight).

For example, use of this system has made it possible to scale down theoptimization problem of the wing box from 20 to 4 variable geometricparameters.

In an exemplary embodiment to described further below, the number ofvariables in optimization of a wing box are decreased. In the case ofthe wing box, the load carrying components are TPM shapes andTPM-panels. TPM-panels are reinforced with stringers or stringers andribs, and vertical walls manufactured out of sheets reinforced withpillars. Following the system tool outlined above, moving from simplecalculation schemes to more complicated ones based on analysis ofdifferent stability failure modes in the wing box structure enablesdivision of wing design into the following structural members: stringer,stringer panel, rib-reinforced stringer panel, pillar-reinforced wallpanel.

Each of the above structural members constitutes a certain stage insolution of the general problem of optimization of the wing box. Thedimensionality of the problem is reduced at each stage.

At the first stage, six variable cross section dimensions of thestringer profile are expressed as a function of the only forcecompressing the stringer and it does not appear as a variable. In thesecond stage of the optimization problem, eight variable parameters,taking into account the results of the first stage, are expressedthrough two variable parameters, the normal stress and the sheetthickness. In the third stage, thirteen variable parameters, taking intoaccount the results of the second stage, are expressed through the sametwo variable parameters, the normal stress and the sheet thickness. Inthe fourth stage, seven variable parameters (independent of the previousparameters) are expressed through two variable parameters, the maximumthickness of the aerodynamic wing profile and the wall thickness. As theend result, the problem of optimization of the thin wall members of theload carrying wing box, comprising twenty variable cross-sectiondimensions parameters, is reduced to four variable parameters, namelythe maximum thickness of the aerodynamic wing profile, sheet thickness,wall thickness, and normal stress. Each of the considered structuralmembers in wing box design optimization is capable of beingindependently employed in practice in optimum design of thin wallstructures.

FIGS. 1-8 show a set of distinct TPM configurations associated with anumber of different variants defined later below.

Likewise, FIGS. 10-15 show various TMP-panel combinations and in somecases from differing perspectives.

Each TPM is defined by at least a main strip (b) and one or moreadditional strips which form web- and flange-type strips. A web (or webstrip) is characterized by two common longitudinal reinforcing ribs,while a flange (or flange strip) is characterized by one commonlongitudinal reinforcing rib and one free longitudinal reinforcing rib.

Referring back to the figures, FIG. 1 shows a TPM (variant TPM I) havinga closed rectangular shape, two main web strips and two additional webstrips in accordance with an exemplary embodiment of the invention.

FIG. 2 shows a TPM (also variant TPM I) with closed triangular shape,two main web strips and one further web strip.

FIG. 3 shows an I-shaped TPM (variant TPM II) which has one main webstrip and four flanges.

FIG. 4 shows a Z-shaped TPM (also variant TPM II) which has one main webstrip and two flanges.

FIG. 5 shows a C-shaped TPM (variant TPM II) with one main web strip andtwo flanges.

FIG. 6 shows a T-shaped TPM (variant TPM II) with one main flange andtwo further flanges.

FIG. 7 shows an L-shaped TPM (variant TPM II) with one main flange andone further flange.

FIG. 8 shows a U-shaped TPM (variant TPM Ill) with two main inclined webstrips, one web strip and two flanges.

FIG. 9 diagrammatically maps shape efficiency factor Σ as a function ofwidth b, which shall be discussed in greater detail below.

FIG. 10 is TPM-panel combination formed by joining an L-shaped TPM to asheet in open cross-section configuration;

FIG. 11 is an inverted perspective of the configuration in FIG. 10showing more clearly the I- and L-shaped TPMs positioning on the panelwhich function to simultaneously reinforce the panel in the longitudinaland transverse directions, respectively.

FIG. 12 is a sectional view of FIG. 11 taken along line “A-A”.

FIG. 13 is a further sectional view as seen in the direction of arrow“B” in FIG. 11.

FIG. 14 is a cross-sectional perspective of a further TPM-panelconfiguration defined by a U-shaped TPM in accordance with a furtherembodiment;

FIG. 15 is a further cross-sectional perspective of a TPM-panelconfiguration substantially as shown in FIG. 14 but including, inaddition to the U-shaped TPM, a further Z-shaped TPM to cooperativelyreinforce the panel in both longitudinal and transverse directions,respectively.

The proposed TPM shape according to the variant I (hereinafter, TPM I)is characterized in that the ratio of the width of the additional stripwith common reinforcing ribs to the width of the main strip satisfiesthe range:

a/b=0.3 to 0.7  (1),

and the ratio of the thickness of additional strip with commonreinforcing ribs to the thickness of the main strip satisfies the range:

δ_(a)/δ_(b)=1.0 to 3.0  (2),

where: a is the width of the additional strip with common reinforcingribs;

b is the width of the main strip;

δ_(a) is the thickness of the additional strip with common reinforcingribs; and

δ_(b) is the thickness of the main strip.

As provided, the stiffness of the main strip does not exceed thestiffness of the additional strip, specifically, δ_(a)/a≥δ_(b)/b.

The proposed TPM according to the variant II (hereinafter, TPM II) ischaracterized in that the ratio of the width of the additional stripwith the free reinforcing rib and the common reinforcing rib to thewidth of the main strip satisfies the range:

c/b=0.05 to 0.3  (3),

and the ratio of the thickness of the additional strip with the freereinforcing rib and the common reinforcing rib to the thickness of themain strip satisfies the range:

δ_(c)/δ_(b)=1.0 to 3.0  (4),

where: b is the width of the main strip;

c is the width of the additional strip with the free reinforcing rib andthe common reinforcing rib;

δ_(b) is the thickness of the main strip; and

δ_(c) is the thickness of the additional strip with the free reinforcingrib and the common reinforcing rib.

As provided, the stiffness of the main strip does not exceed thestiffness of the additional strip, specifically, δ_(c)/c≥δ_(b)/b.

The proposed TPM according to the variant III (hereinafter, TPM III) ischaracterized in that the ratio of the width of the additional stripwith common reinforcing ribs to the width of the main strip satisfiesthe range:

a/b=0.3 to 0.7  (1),

and the ratio of the thickness of the additional strip with commonreinforcing ribs to the thickness of the main strip satisfies the range:

δ_(a)/δ_(b)=1.0 to 3.0:  (2),

with this, the ratio of width of the additional strip with the freereinforcing rib and the common reinforcing rib to the width of the mainstrip satisfies the range:

c/b=0.05 to 0.3  (3),

and the ratio of the thickness of the additional strip with the freereinforcing rib and the common reinforcing rib to the thickness of themain strip satisfies the range:

δ_(c)/δ_(b)=1.0 to 3.0  (4),

where: a is the width of the additional strip with common reinforcingribs;b is the width of the main strip;

c is the width of the additional strip with the free reinforcing rib andthe common reinforcing rib;

δ_(a) is the thickness of the additional strip with common reinforcingribs;

δ_(b) is the thickness of the main strip;

δ_(c) is the thickness of the additional strip with the free reinforcingrib and the common reinforcing rib;

with this, the stiffness of the main strip does not exceed the stiffnessof the additional strip; specifically, δ_(a)/a≥δ_(b)/b andδ_(c)/c≥δ_(b)/b.

Besides, in this variant of the TPM, the stiffness of the additionalstrip with common reinforcing ribs does not exceed the stiffness of theadditional strip with the free reinforcing rib and the commonreinforcing rib; specifically, δ_(a)/a≥δ_(b)/b and δ_(c)/c≥δ_(b)/b,which follows from expressions (1), (2), (3) and (4).

The second type of subject of the group of inventions is two variants ofTPM-panel configurations, IV and V (hereinafter, TPM-panel IV andTPM-panel V, or simply panel IV and panel V), based on all said variantsof TPM, I, II and III.

The technological result of the panel according to the variant IV isachieved by that the panel comprises a sheet and a number of TPMs,having relations between the shape dimensions complying with theexpressions in equations (1)-(4) above, which are installed across itswidth longitudinally with even pitch; with this, the main strip(s) andadditional strip(s) are forming with the sheet an open cross-sectionconfiguration, wherein the thickness of the sheet of the panel accordingto the invention satisfies the following expression:

δ=(0.0006 to 0.0035)l  (5),

and the pitch of the longitudinal installation satisfies the expression:

b _(c)=(20 to 65)δ  (6),

where: δ is the thickness of the sheet of the panel;

l is the length of the thin wall profile member forming with the sheetthe open cross-section configuration; and

b_(c) is the pitch of the longitudinal installation of thin wall profilemembers forming with the sheet the open cross-section configuration.

The stiffness of the main strip does not exceed the stiffness of theadditional strip, and the stiffness of the additional strip with commonreinforcing ribs does not exceed the stiffness of the additional stripwith the free reinforcing rib and the common reinforcing rib;specifically, δ_(b)/b≤δ_(a)/a≤δ_(c)/c.

Besides, the panel IV can be additionally equipped with the TPMinstalled transversally and having the above relations of the shapedimensions (1)-(4). With this, it is expedient to install these TPM withthe pitch of transversal installation

l _(n)=(10 to 60)b _(c)  (7),

where: l_(n) is the pitch of transversal installation of thin wallprofile member for the case of longitudinally installed thin wallprofile member forming with the sheet the open cross-sectionconfiguration;

b_(c) is the pitch of the longitudinal installation of thin wall profilemembers forming with the sheet the open cross-section configuration.

The technological result of panel V is achieved by that the panelcomprises a sheet and a number of TPMs, having relations between theshape dimensions complying with the expressions in equations (1)-(4),which are installed across its width longitudinally with even pitch;with this, the main strip(s) and additional strip(s) are joined with thesheet to form a closed cross-section configuration, wherein thethickness of the sheet of the panel according to the invention satisfiesthe following expression:

δ=(0.0006 to 0.0035)l  (5)

and the pitch of the longitudinal installation satisfies the expression:

b ¹ _(c)=(40 to 130)δ  (8),

where: δ is the thickness of the sheet of the panel;

l is the length of the thin wall profile members forming with the sheetthe closed cross-section configuration; and

b¹ _(c) is the pitch of longitudinal installation of thin wall profilemembers forming with the sheet the closed cross-section configuration.

The stiffness of the main strip does not exceed the stiffness of theadditional strip, and the stiffness of the additional strip with commonreinforcing ribs does not exceed the stiffness of the additional stripwith the free reinforcing rib and the common reinforcing rib,specifically, δ_(b)/b≤δ_(a)/a≤δ_(c)/c.

Besides, the panel according to the variant V can be additionallyequipped with the TPM installed transversally and having the aboverelations of the shape dimensions (1) to (4). With this, it is expedientto install these TPM with the pitch of transversal installation

l ¹ _(n)=(1.5 to 10)b ¹ _(c)  (9),

where:

l¹ _(n) is the pitch of transversal installation of thin wall profilemembers for the case of longitudinally installed thin wall profilemember farming with the sheet the closed cross-section configuration;and

b¹ _(c) is the pitch of the longitudinal installation of thin wallprofile members forming with the sheet the closed cross-sectionconfiguration.

Thus, to achieve an optimum weight TPM for a given set of constructiverestrictions the process to be followed involves first providing a TPMhaving a cross-section that includes at least one of (1) at least twomain strips and at least one additional strip having ends connectingwith respective ends of two of the at least two main strips andselecting dimensions such that each main strip has a thickness δ_(b) anda width b and the additional strip has a thickness δ_(a) and a width aso that δ_(b)/b is not larger than δ_(a)/a, and (2) at least one mainstrip and at least one additional strip having one end connecting withan end of the main strip and selecting dimensions such that the mainstrip has a thickness δ_(b) and a width b and the additional strip has athickness δ_(c) and a width c and so that δ_(b)/b is not larger thanδ_(c)/c.

The next step involves selecting set of cross-section dimensions ratiosvalues within established ranges values. A set of constructiverestrictions is established that map to actual cross-section dimensions.

Then, based on the set of ratios values and constructive restrictions, arespective set shape efficiency factors values Σ₁, Σ₂ . . . Σ_(n), aredetermined wherein each of the shape efficiency factor values is definedby:

Σ=K _(f) ·K _(m), where

K_(f)=(i²/F)^(2/5) is an overall stability factor,

K_(m)=K^(1/5)/(b/δ_(b))^(2/5) is a local stability factor,

b, δ_(b) are the width and the thickness of the main strip,respectively,

i, F are the radius of gyration and the cross-section area,respectively, and

K is the coefficient in the known formula for local stability stress.

In a subsequent step, a maximum of the shape efficiency factor valueΣ_(max) is determined from within the respective set of determined shapeefficiency factors values Σ₁, Σ₂ . . . Σ_(n).

Then based on this determination of the shape efficiency factor maximumvalue Σ_(max), a next step is to ascertain TPM ratio values.

From this, a TPM pattern having cross-section dimensions and ratiovalues which result in a maximum shape efficiency factor value Σ_(max)is identified. Implicitly, this same pattern also ensures the reliableoperation and weight-optimized result of a TPM for the given set ofdesired, predefined constructive restrictions.

The proposed approach calculates maximum shape efficiency factor valuesfor different shape TPMs. Once this is done, an overall maximum shapeefficiency factor value Σ_(0max) is identified as well as its associatedshape. This shape determines among all TPM shapes the best TPMconfiguration to employ.

As previously explained, a TPM could be combined with two or morestrips. In such cases, if buckling were to occur, it is most likely tooccur first in the strips of the TPM. We can refer to this as “localfailure mode” and is characterized by warp along one or morecross-sections in addition to any stresses produced linearly in relationto the position of common longitudinal ribs of the strips.

This local failure mode may result in overall buckling failure mode andcollapse of the entire TPM subject to compression forces. This in turncould lead or contribute to overall TPM-panel combination failure, whichcould result in severe system or device failure.

Optimization, therefore, as contemplated herein involves applying thetechniques proposed to determine possible failure condition along anumber of cross-sections of the TPM-panel combination.

The presently proposed approach therefore is a design optimizationtechnique that algorithmically looks at alternative TPM and TPM-panelcombinations, and on the basis of predefined constructive restrictions(structural, material and fabricated requirements) identifies acombination of variables from which cross-section dimensions aredetermined. By making incremental changes to cross-section dimensions,and for a given pattern, determining possible failure conditions, it ispossible to arrive at a best result. This result it has been determinedis a much more efficient and cost effective way to select a best patternthan has been done using conventional (usually iterative) techniques.

This next section will now set out to describe application of thesuggested techniques to arrive at somewhat more complicated patternsinvolving longitudinal and transversal reinforcing TPM-panelcombinations IV and V, respectively. Here again the goal is to minimizepanel weight with respect to a given set of constructive restrictions.

TPM-panel combinations IV and V are “wide column” structures, whereasTPMs I-III are all simple or “narrow column” structures, for purposes ofdiscussion. Wide and narrow is used here to describe stability relatedattributes in overall and local terms. A configuration cross-section is“wide”, for instance, when there is both TPM and panel simultaneouslybeing subjected, at that cross-section, to various load-bearingstresses. In case of “narrow” design, only the TPM cross-section isconsidered to bear load stress.

“Local” is meant to refer to a “localized failure condition”, i.e., acondition that is limited to the TPM (or TPM-panel combination) only.When the failure condition expands beyond the TPM (or TPM-panel) thenthere is said to be an “overall failure condition”.

In the case of a TPM-panel combination, the sheet (which serves as thepanel) provides substantial additional stability to the combination,which is one reason why it is provided. When referring to localstability, the benefit in stability of the sheet is not accounted for.However, in calculating overall stability, then the contribution instability by the sheet is accounted for through appropriate calculationsdiscussed below. This is achieved by treating wide cross-sections thesame as narrow cross-sections when performing calculations. The aim isto arrive at a shape efficiency factor that is relevant.

Referring again to the variant TPM configurations shown in FIGS. 1 to 8,cross-section dimensions selection of each TMP will next be described byapplication of a set of ratios rules.

As should be appreciated, a TPM is configured to structurally beimpacted when a compressive load P is applied. The following TPMconfigurations have been shown: closed rectangle (FIG. 1), closedtriangle (FIG. 2), I-shaped (FIG. 3), Z-shaped (FIG. 4), C-shaped (FIG.5), T-shaped (FIG. 6), L-shaped (FIG. 7), and U-shaped (FIG. 8).

A TPM may comprise one or more main web strips 2 (as shown in FIGS. 1-5and 8); or one or more main flanges 3 (as shown in FIG. 6 and FIG. 7).Each TPM may further include one or more main strip(s) 4 characterizedby two common longitudinal reinforcing ribs or one free longitudinalreinforcing rib and one common longitudinal reinforcing rib 5,respectively, as shown in FIGS. 10-15. Additional flange(s) 6 (shown inFIGS. 3 to 8) and web strip 7 (shown in FIGS. 1, 2 and 8) may beincluded and which are defined by (i) a width which is less than that ofmain strip 4 and (ii) a thickness not less than that of main strip 4.

Further rules that should be followed are set forth below.

For example, the stiffness of main strip 4 should not exceed that of theadditional strip (flanges 6, web strips 7): specifically,δ_(a)/a≥δ_(b)/b and δ_(c)/c≥δ_(b)/b. The stiffness of the additionalstrip with two common longitudinal reinforcing ribs, web 7 (FIG. 8),should not exceed the stiffness of the additional strip with one freelongitudinal reinforcing rib and one common longitudinal reinforcingrib, flange 6 (FIG. 8): specifically, δ_(a)/a≤δ_(c)/c.

The additional flange 6 or the additional web 7 can be located withrespect to main strip 4 either at a 90° angle as in FIGS. 1, 3 to 7, orat a different angle as in FIGS. 2 and 8.

Width and thickness of main webs 2, flanges 3 and additional webs 7,flanges 6 in the cross-sections of TPM (FIGS. 1 to 8) should satisfyexpressions (1), (3), (10):

a/b=0.3 to 0.7  (1),

c/b=0.05 to 0.3  (3),

δ_(a)/δ_(b)=δ_(c) /b=1.0 to 3.0  (10),

where: a, b, c, δ_(a), δ_(b), and δ_(c) are, respectively, width andthickness of the additional web, the main web or flange and theadditional flange.

For example, width a of additional web 7 has its length measured along across-section medial line (the line equidistant from longitudinal edgelines of cross-section) of the web 7 between the respective lines ofmain webs 2 adjacent to the web 7 (FIGS. 1, 2 and 8).

Width b of main web 2 or flange 3 has its length measured along themedial line of the cross-section of main web 2 or flange 3 between therespective lines of adjacent strips (FIGS. 1 to 8).

Width c of additional flange 6 has its length measured along the medialline of the cross-section of flange 6 from the medial line of main web 2or flange 3 to the free longitudinal reinforcing rib of additionalflange 6 (FIGS. 3 to 8).

Thickness δ_(a), δ_(b), and δ_(c) corresponding to dimensions a, b, c isdefined as the distance measured between the edges of cross-sections ofwebs and flanges.

The above expressions (1), (3), (10) should be applied where variant TPMI, II or III are employed with following shapes: I-shape with one mainweb 2 and four additional flanges 6 (FIG. 3); Z-shape (FIG. 4) with twoadditional flanges 6 located both sides from the main web 2; or C-shapewith additional flanges 6 located the same side of the main web 2 (FIG.5); or U-shape with main webs 2, additional web 7 and additional flanges6 (FIG. 8); or T (FIG. 6) and L (FIG. 7)—with the main flange 3 andadditional flange(s) 6; or closed rectangular (FIG. 1) and triangular(FIG. 2); as well as shapes with other arrangement and quantity of mainand additional webs and flanges.

The ratio range values of widths and ratios of thicknesses of main webs2 and flanges 3, additional flanges 6 and webs 7 are obtained using thegeneralizing parameter with various shapes of TPM. Reference was made tothis earlier when referring to a “shape efficiency factor Σ, where

Σ=K _(f) ˜K _(m), where:

K_(f)=(i²/F)^(2/5) is an overall stability factor,

K_(m)=K^(1/5)/(b/δ_(b))^(2/5) is a local stability factor,

b, δ_(b) are the width and the thickness of the main web 2 or flange 3,respectively;

i, F are the radius of gyration and the area of shape of any TPM shownin FIGS. 1 to 8, respectively; andK is the coefficient in the known formula for local stability criticalstress, depending on ratios (1), (3), (10) of TPM shape dimensions.

TPMs may be compared in terms of weight as well. The greater the maximumvalue of factor Σ for a particular shape, the less its TPM weight.

At the same time, within the specified ranges, maintaining the values ofthe above ratios, variation of shape absolute (actual) dimensions ispossible which enables to provide for constructive restrictions notentailing a considerable increase of the weight of TPM. Beyond theseranges, the weight of TPM increases significantly.

The graphic illustration of the shape efficiency factor Σ versus thewidth b of the main strip is shown in FIG. 9. As can be seen from thisplot, the factor Σ possesses, for each shape, a maximum value. For eachTPM shape, a corresponding shape efficiency factor maximum value exitsthat falls inside a range of ratio values.

FIG. 9 therefore illustrates the significance of a shape efficiencyfactor. The wider a structural shape cross-sectional material is“distributed” (i.e., placed at the farthest distance from the neutralaxis), the bigger the value of the overall stability factor K_(f) andits overall stability.

At the same time, the local stability factor K_(m) that characterizeslocal stability decreases. As a result, when functions of overall andlocal stability have equal values, the function of the shape efficiencyfactor has its maximum value (this being different for different TPMcross-section shapes). Furthermore, this maximum value ensures theminimum TPM shape cross-section area (and hence the minimum weight).

Different cross-section shapes can also be compared by their associatedweight values. The bigger the maximal value of the shape efficiencyfactor Σ of a particular shape, the smaller its associated TPM shapeweight value.

At the same time, it is possible to maintain, within specified ranges,the values of the above ratios through a variation of shape actual(absolute) cross-section dimensions. This has the benefit of, for agiven set of constructive restrictions, being able to select actualcross-section dimensions without compromising considerably TPM weight.Inside the ranges, for each TPM shape, the values of the shapeefficiency factor have insignificant deviation from its theoreticalmaximal values and thus insignificant deviation from its theoreticalminimum weight values. Outside the ranges, the shape efficiency factorvalues decrease significantly, so the weight of the TPM increasessignificantly.

Having identified that a range of values for shape efficiency factorexists, it is not only possibly to select a best design TPM for a givenapplication, it is now also possible to create many new TPMs whichlikewise are defined by a shape efficiency factor that varies onlyslightly within a desired range and with respect to weight, and muchmore significantly outside the range.

To minimize weight during design development, the idea is to use aprogramming tool that algorithmically reduces the design choice down toa best TPM selection based on predefined, or otherwise known,cross-section dimension ratio values falling with a desired range.

As a design is tweaked to meet other design constraints, the tool willrecognize when certain cross-section dimension ratios appear to havefallen outside a desired range of values and prompt the designer toselect a different TPM, or change the material of the TPM. In the latterinstance, a stronger material will necessarily change the shapeefficiency factor for a TPM of a given shape.

The above-described principles are equally relevant when designingTPM-panel combinations.

FIG. 10 shows a TPM-panel combination of variant IV. This variant isbased on the L-shaped variant TPM II described above and shown in FIG.7. In a fully-deployed state, a TPM-panel will be subjected primarily toa compressive load q directed along a length I that's distributed acrossa width B of the TPM-panel. The TPM-panel includes sheet 8 and a varianttype TPM II attached to one of the sides 9 of this sheet 8 across itswidth and installed longitudinally with even pitch. Multiple TPMs(variant II) are shown. Each TPM is integrably coupled to sheet 8 in anopen cross-section configuration. Each TPM II is an L-shape type TPM andconsists of main flange 3 and one additional flange 6.

Width b and c of main flanges 3 and of additional flanges 6 and thethickness of these, δ_(b) and δ_(c), respectively, satisfy theexpressions:

c/b=0.05 to 0.3  (3),

δ_(c)/δ_(b)=1.0 to 3.0  (4).

The other dimensions of TPM-panel (variant IV) cross-section in FIG. 10satisfy the following expressions:

δ=(0.0006 to 0.0035)l  (5),

b _(c)=(20 to 65)δ  (6),

where: δ is the thickness of sheet 8;

b_(c) is the pitch of the longitudinal installation of the TMPs restingon sheet 8 in the open cross-section configuration shown and described;and

l is the length of TPM II.

As provided, the stiffness of the main strip does not exceed thestiffness of the additional strip, specifically, δ_(c)/c≥δ_(b)/b.

FIG. 14 shows a TPM-panel (variant V) which is based on a TPM of variantIII of U-shape (as in FIG. 8). Here, the TMP is formed with sheet 8 inclosed cross-section configuration. Each TPM III is characterized bymain webs 2, additional web 7 and additional flanges 6.

The values of width a, b, c of additional webs 7, main webs 2, andadditional flanges 6 and values of thickness corresponding to thesedimensions, δ_(a), δ_(b), δ_(c) of the shape of TPM III of the TPM-panelV satisfy the following expressions:

a/b=0.3 to 0.7  (1),

c/b=0.05 to 0.3  (3),

δ_(a)/δ_(b)=δ_(c) /b=1.0 to 3.0  (10).

The other dimensions of the panel V cross-section (FIG. 14) satisfy thefollowing expressions:

δ=(0.0006 to 0.0035)l  (5),

b ¹ _(c)=(40 to 130)δ  (8),

where:

δ is the thickness of the sheet 8;

b¹ _(c) is the pitch of the longitudinal installation of thin wallprofile members forming with the sheet 8 the closed cross-sectionconfiguration; and

l is the length of TPM III.

The stiffness of the main strip should not exceed the stiffness of theadditional strip, and the stiffness of the additional strip with commonreinforcing ribs should not exceed the stiffness of the additional stripwith the free reinforcing rib and the common reinforcing rib,specifically, δ_(b)/b≤δ_(a)/a≤δ_(c)/c.

To reinforce TPM-panels IV and V in longitudinal direction, additionalTPMs of variant I, II and III are shown embodied. One skilled in the artwould appreciate that this is done only for discussion purposes, andthat other TMP shapes, including for example, Z-, T-, C-, rectangular,or triangular shapes.

The TPM-panels IV and V, shown in FIGS. 10 and 14 function as follows.

The distributed compressive load q is reacted by both TPM I, II, III andthe sheet of TPM-panel 8. The load-bearing capacity of the TPM-panelsIV, V is provided for by virtue of selection of optimum dimensions ofcross-section of the panel: dimensions of shapes of TPM are selectedbasing on the expressions of equations (1), (3), (4) and (10); thethickness of sheet 8 and the pitch of TPM longitudinal installation thathave been selected based on rules satisfying the expressions inequations (5), (6) and (8), respectively.

FIGS. 11-13 and 15 show panels IV, V with TPM II, III installed bothlongitudinally and transversally.

FIGS. 11 to 13 show panel IV based on longitudinally installed I-shapedTPM II and transversally installed L-shaped TPM II.

In its course of operation, a TPM-panel IV will reacts primarily to anycompressive load q directed along length L of any longitudinallyinstalled TPM (see pos. 11 in FIGS. 11 and 12). These same TPMs areattached to one of the sides 12 of sheet 10 across its width and formedintegrally with sheet 10 in open cross-section configuration.

TPM 13 are installed transversally with even pitch across the length ofthe same side 12 of sheet 10 and are embodied with cut-outs 14 in whichthe longitudinally installed TPM 11 are located (FIGS. 11-13). TPM 11possesses the main web 15 embodied as the main strip 16; at each of itsreinforcing ribs 17, an additional flange 18 is formed with the widthless than that of the strip 16 and with thickness not less than that ofthe strip 16 (FIG. 12).

The width b and c of main webs 15 and additional flanges 18, andthickness corresponding to these dimensions, δ_(b), δ_(c), respectively,of the shape of longitudinally installed TPM 11 satisfy the expressions:

c/b=0.05 to 0.3  (3),

δ_(c)/δ_(b)=1.0 to 3.0  (4),

With this, the stiffness of the main strip does not exceed the stiffnessof the additional strip, specifically, δ_(c)/c≥δ_(b)/b.

Each transversally installed TPM 13 is embodied as an L-shape andpossesses a main flange 19 embodied as a main strip 20 across the widthof which, at one of its longitudinal reinforcing ribs 21, an additionalflange 22 is formed (FIG. 13).

Relations of shape dimensions of TPM 13 should also satisfy theexpressions (3) and (4).

The other dimensions of the cross-section of TPM-panel IV should satisfythe expressions:

δ=(0.0006 to 0.0035)L/n _(n)  (5),

b _(c)=(20 to 65)δ  (6),

where:

L is the length of the panel;

n_(n) is the number of transversally installed TPM 13;

δ is the thickness of the sheet 10;

b_(c) is the pitch of the longitudinal installation of TPM 11 formingwith the sheet 10 the open cross-section configuration. In the panel IVwith longitudinally installed TPM 11 of I-shape forming with the sheet10 the open cross-section configuration, the pitch l_(n) of thetransversal installation of L-shaped TPM 13 (shown in FIG. 11),satisfies the expression:

l _(n)=(10 to 60)b _(c)  (7).

FIG. 15 shows the TPM-panel V based on the longitudinally andtransversally installed members 11 and 13, respectively. Thelongitudinally installed TPM 11 is embodied as a U-shape and formed withsheet 10 in closed cross-section configuration. The transversallyinstalled member 13 is of a Z-shape.

The width a, b, c of additional webs, main webs, and additional flangesand thickness corresponding to these dimensions δ_(a), δ_(b), δ_(c) ofthe longitudinally and transversally installed TPM 11, 13 of theTPM-panel V shown in FIG. 15 satisfy the following expressions:

a/b=0.3 to 0.7  (1),

c/b=0.05 to 0.3  (3),

δ_(a)/δ_(b)=δ_(c)/δ_(b)=10.0 to 3.0  (10).

The stiffness of the main strip should not exceed the stiffness of theadditional strip, and the stiffness of the additional strip with commonreinforcing ribs should not exceed the stiffness of the additional stripwith the free reinforcing rib and the common reinforcing, specifically,δ_(b)/b≤δ_(a)/a≤δ_(c)/c.

The other cross-section dimensions of the panel V (FIG. 15) shouldsatisfy the expressions:

δ=(0.0006 to 0.0035)L/n ¹ _(n)  (5),

b ¹ _(c)=(40 to 130)δ  (8),

where: δ is the thickness of sheet 10;

b¹ _(c) is the pitch of installation of U-shaped TPM 11 formed withsheet 10 in closed cross-section configuration;

L is the length of the panel; and

n¹ _(n) is the number of transversally installed TPMs 13.

The pitch l¹ _(n) of transversal installation of TPM 13 of TPM-panel Vprovided in the longitudinal installation of TPM 11 and formed withsheet 10 in closed cross-section configuration, should further satisfythe expression:

l ¹ _(n)=(1.5 to 10)b ¹ _(c)  (9).

The width B of TPM-panels IV, V shown in FIGS. 10, 11, 14 and 15 isdetermined from the following expressions:

B=n _(c) ·b _(c) =n ¹ _(c) ·b ¹ _(c)  (11),

where n_(c), n¹ _(c) are the numbers of longitudinally installed TPM ofTPM-panels IV, V forming with the sheet the open and closedcross-section configuration, respectively.

The length L of TPM-panels IV, V shown in FIGS. 11 and 15 may bedetermined from the following expressions:

L=n _(n) ·l _(n) =n ¹ _(n) ·l _(n)  (12),

where n_(n) and n¹ _(n) are the number of transversally installed TPMsfor those longitudinal TPMs formed with the sheet in open and closedcross-section configurations, respectively.

For transversal and longitudinal reinforcing of TPM-panels IV and V, TPMof other shapes can also be employed.

In the course of operation of panels IV and V under the compressive loadq, the load-bearing capacity of the panel is provided for by virtue ofselection of the optimum dimensions of cross-sections of longitudinallyand transversally installed TPM, thickness of the sheet, pitches oflongitudinal and transversal installation corresponding to the minimumweight of the panel.

The presented examples and drawings illustrated herein are provided onlyby way of example. The shapes and configurations were selected due totheir wide use in applications where thin wall profile structures aredesirable. The illustrations are not meant to in any way limit the scopeand spirit of application of the variants of the present invention.While a number of different variants of TMP-panel combinations have beenshown and described, this too is not meant to be equally limited in anyway.

It should further be stated that while TPM-panels are not as commonlyused as simple TPM configurations, the mathematical relations expressedby (1), (3) and (10) are equally relevant to TPM-panels as they are toTMP-only configurations.

Thus, to achieve an optimum weight TPM for a given set of constructiverestrictions the process to be followed involves first providing a TPMhaving a cross-section that includes at least one of (1) at least twomain strips and at least one additional strip having ends connectingwith respective ends of two of the at least two main strips andselecting dimensions such that each main strip has a thickness δ_(b) anda width b and the additional strip has a thickness δ_(a) and a width aso that δ_(b)/b is not larger than δ_(a)/a, and (2) at least one mainstrip and at least one additional strip having one end connecting withan end of the main strip and selecting dimensions such that the mainstrip has a thickness δ_(b) and a width b and the additional strip has athickness δ_(c) and a width c and so that δ_(b)/b is not larger thanS/c.

The next step involves selecting set of cross-section dimensions ratiosvalues within established ranges values. A set of constructiverestrictions is established that map to actual cross-section dimensions.

Then, based on the set of ratios values and constructive restrictions, arespective set shape efficiency factors values Σ₁, Σ₂ . . . Σ_(n), aredetermined wherein each of the shape efficiency factor values is definedby:

Σ=K _(f) ·K _(m), where

K_(f)=(i²/F)^(2/5) is an overall stability factor,

K_(m)=K^(1/5)/(b/δ_(b))^(2/5) is a local stability factor,

b, δ_(b) are the width and the thickness of the main strip,respectively,

i, F are the radius of gyration and the cross-section area,respectively, and

K is the coefficient in the known formula for local stability stress.

In a subsequent step, a maximum of the shape efficiency factor valueΣ_(max) is determined from within the respective set of determined shapeefficiency factors values Σ₁, Σ₂ . . . Σ_(n).

Then based on this determination of the shape efficiency factor maximumvalue Σ_(max), a next step is to ascertain TPM ratio values.

From this, a TPM pattern having cross-section dimensions and ratiovalues which result in a maximum shape efficiency factor value Σ_(max)is identified. Implicitly, this same pattern also ensures the reliableoperation and weight-optimized result of a TPM for the given set ofdesired, predefined constructive restrictions.

The proposed approach calculates maximum shape efficiency factor valuesfor different shape TPMs. Once this is done, an overall maximum shapeefficiency factor value Σ_(0max) is identified as well as its associatedshape. This shape determines among all TPM shapes the best TPMconfiguration to employ.

TPM Computational Analysis—High Level Operational Flow

FIG. 16 is a high-level flow diagram 100 describing the operation flowof a model and analysis TPM design tool configured for selecting anoptimum weight TPM in accordance with an exemplary embodiment.

The proposed process of designing TPMs is facilitated by a model andanalysis design tool. The tool performs computational analysis onrelevant input variables. In this regard, we will functionally delineateprocess functions into an input stage and a computation stage.

The input stage, as the name implies, involves the designer inputtinginto the tool relevant design parameters (step 120). The relevant designparameters may include preset parameters such as concentratedcompressive force value, material properties (compression yield stressvalue, compression proportionality limit value, tensile ultimate stressvalue, compression limiting stress value, compression normal modulusvalue), pattern of axes, length of the TPM under analysis, and otherrelevant parameters.

The tool uses the received parameters to establish weight minimizationparameters which uses the received parameters to establish weightminimization parameters which include consideration of constructiverestrictions (including fabricated restrictions) to TPM cross-sectiondimensions (step 130). Finally, the designer is asked to select from setof TPM cross-section dimension ratio values, respecting fabricationrequirements of previous step, within pre-established TPM range values(step 140).

Having received all relevant input date, the tool must process the setof ratio values, respecting constructive restrictions, to obtaincorresponding set of shape efficiency factor values (step 150).

The processing in the previous step results in identification of optimumratio values according to maximum shape efficiency factor value (step160).

The tool then calculates a maximum stress value using the maximum shapeefficiency factor value to obtain maximum stress value (step 170). Atstep 180, the tool calculates optimum cross-section dimension valuesusing maximum stress value and optimum ratio values.

TPM-Panel Computational Analysis—High Level Operational Flow

FIG. 17 is a high-level flow diagram 200 describing the operation flowof a model and analysis TPM design tool configured for selecting anoptimum weight TPM-panel in accordance with a further exemplaryembodiment.

Referring to FIG. 17, a first step in design optimization of TPM-panelcombinations involves receiving preset panel uniform compressive forcevalue; panel material properties (TPM and sheet)—compression yieldstress value, compression proportionality limit value, tensile ultimatestress value, compression limiting stress value, compression normalmodulus value; pattern of axes (TPM shape); and panel length and width(step 210). Any constructive (including fabricated) restrictions toTPM-panel including minimum feasible sheet thickness value areestablished and entered into the tool to be used in its calculations(step 220). Steps 210 and 220 are part of the input stage functionsperformed by the tool.

Once the information has been collected the tool is ready to beginprocessing this information. This is performed by a calculations stage.

The first calculation is to generate a maximum panel stress value (step230), followed by generating TPM concentrated compressive force value(step 240). This then is followed by generating minimum weightcross-section dimensions values using the methodology described inoperational flow diagram 100 (step 250). The tool then calculates andgenerates an optimum pitch value of longitudinally installed TPMs (step260). In the final step 270, the optimum pitch value of transversalinstallation TPMS is obtained.

The solution of the panel optimization problem is based on the existingmethods of direct calculations of its strength, as well as on theresults obtained in solution of the problem of the optimum compressedTPM.

The panel is considered as a “wide column” losing stabilitysimultaneously in overall and local modes. It is assumed in the practiceof TPM-reinforced panels that the link of TPM with skin is pivotal. Inthis assumption, in the local stability calculations, the mutual effectof TPM and skin is not accounted for, while in the overall stabilitycalculations the effect of the skin is accounted for through theeffective bending rigidity of the TPM. In the calculation schemeselected, the shape efficiency factor for a “wide column” shall beassumed the same as for the “simple (narrow) column”.

For the example, the panel with Z-shape TPM (forming with the sheet anopen configuration) shall be considered. For the panel with the U-shapedTPM (forming with the sheet a closed configuration), the final resultsare presented. The results of calculations are presented for panels withU-shaped, Z-shaped and L-shaped TPM.

To summarize, the computer-implemented methodology in accordance with anexemplary embodiment, involves the sequence of steps of:

(i) identifying constructive restrictions associated with the panelsheet;(ii) calculating a maximum panel stress value on the basis of theconstructive restrictions;(iii) calculating a TPM concentrated force value;(iv) calculating for any TPM that is to be longitudinally disposed onthe panel sheet, and for a first corresponding set of constructiverestrictions, a first set of optimum cross-section dimensional ratiovalues;(v) calculating for any TPM that is to be transversally disposed on thepanel sheet, and for a second corresponding set of constructiverestrictions, a second set of optimum cross-section dimensional values;(vi) calculating a first optimum pitch value to pitch position the TPMsto be longitudinally disposed on the panel sheet; and(vii) calculating a second optimum pitch value to pitch position theTPMs to be longitudinally disposed on the panel sheet.

As explained, the constructive restrictions associated with the panelsheet include a minimum feasible sheet thickness value, as well as atleast one of a preset panel uniform compressive force value, panelmaterial properties, TPM pattern of axes, and a length and width ofpanel.

Furthermore, the calculating of at least one of the first and secondsets of optimum cross-section dimensional values of a corresponding TPMinvolves identifying a set of TPM cross-section dimensional ratiovalues, and calculating from a set of inter-dependent parameters,including the set of TPM cross-section dimensional ratio values, theoptimum cross-section dimensional ratio values of the TPM.

The set of inter-dependent parameters includes the corresponding one ofthe set of first and second constructive restrictions. Each of the firstand second sets of constructive restrictions is determined from at leastone of a preset concentrated compressive force value, materialproperties, pattern of axes, and a length of the associated TPM.

Furthermore, the calculating of the optimum cross-section dimensionsratio values of the TPM, involves:

(i) calculating a corresponding set of shape efficiency factor valuesfor the set of TPM cross-section dimensions ratio values, using thecorresponding one of the first and second sets of constructiverestrictions;(ii) calculating a maximum shape efficiency factor value; and(iii) calculating optimum ratio values and a maximum stress value on thebasis of the maximum shape efficiency factor value.

A further example of generating an optimum TPM design given a set ofconstructive restrictions utilizing the more common Z and I shapes, isnow presented.

A Z shape is shown in FIG. 4 while an I shape is shown in FIG. 5.Consider these same TPMs with four variable shape dimensions b, δ_(b),c, and δ_(c). We first introduce relevant ratios:

(i) the ratio of the width of the additional strip with the free rib andthe common rib to the width of the main strip: a₁=c/b, and

(ii) the ratio of the thickness of the additional strip with the freerib and the common rib to the thickness of the main strip:a₂=δ_(c)/δ_(b).

We then express the shape efficiency factor as a function of the tworatios a₁, a₂ and coefficient K=K (a₁, a₂).

The theoretical maximum value of Z shape efficiency factor Σ_(max)=0.538is obtained at the optimum ratio values at a₁ ^(op)=0.15 and at a₂^(op)=2, then the factor K^(op)=6.0.

The theoretical maximum value of I shape efficiency factor Σ_(max)=0.556is obtained at the optimum ratio values at a₁ ^(op)=0.05 and at a₂^(op)=2.5, then the factor K^(op)=6.2.

Feasible ratios for Z shape taking into account constructiverestrictions are presented below.

For Z shape, (at Σ_(max)=0.538) an optimum value of a₁ ^(op)=0.15. Thelength of flange c at this optimum value is not sufficient to be able tosuccessfully mount a rivet (c<2 cm). If the optimum thickness ratiovalue is increased to a₂ ^(op)=2, thickness δ_(b) may prove too thin(less than 1 mm). This makes it difficult to manufacture such a profilein mass-scale production. The most suitable for production is a constantthickness shape with a₂ ^(op)=1, but at some loads and material thevalue flange thickness δ_(c) may be less than the value of the thicknesspanel skin, and hence not recommended.

Given a different set of constructive restrictions, whereby a₁=0.05-0.3and a₂=1.0-3.0, then in this scenario, a Z-shaped TPM may very well be abest option.

Going back to the previous example, it was explained that for a rivet tofit, an a₁ ^(op)=0.3 is desirable. With this, the maximum value Σ=0.515,at a₂ ^(op)=2, therefore Σ=0.515 will be less Σ_(max)=0.538 and weightof this variant Z shape will be more than for Z shape at theoreticalvalue Σ_(max)=0.538 over 4.3% only. In the second example, again for arivet could fit, we will assume that a₁ ^(op)=0.3, and that the mostsuitable for production is a constant thickness profile. Now assuming a₂^(op)=1, the resultant shape efficiency factor is only Σ=0.50. So, bychanging slightly the design configurations the designer has to choosefrom, it is possible to address the problem of being able to provideclearance and thickness for the mounting of a rivet, at an additionalmarginal weight “cost” or “penalty” which is only 5.3% above anon-suitable for riveting configuration, but otherwise lighter inweight.

Beyond specifying ranges, the value of shape efficiency factor coulddecrease significantly. For example assuming values of (a₁ ^(op)>0.3 anda₂ ^(op)<1.0), which would allow fitting of a rivet, and assuming nextthat a₁ ^(op)=0.4 and that thickness δ_(b) can't prove too thin, it isassumed that a₂ ^(op)=0.5, corresponding value Σ=0.425, therefore weightof this variant profile will be more over 21.4% (comparing variant withtheoretical value Σ_(max)=0.538).

Taking this a step further, with a₁ ^(op)>0.3 and a₂ ^(op)>3.0), againso a rivet could fit, and assuming that a₁ ^(op)=0.4 and that the valueflange thickness δ_(c) should be more than the value of the thicknesspanel skin, and if we assume a₂ ^(op)=3.5, we arrive at a resultantΣ=0.367. At this Σ, a substantial weight increase of over 31.8 percentis realized.

Feasible ratios values and corresponding shape efficiency factor valuesfor a Z-shaped TPM as a function of a given set of constructiverestrictions is presented in Table 1 below.

TABLE 1 (Σ_(max) = 0.538) a₁ ^(op) a₂ ^(op) Σ % 1. 0.30 2.0 0.515 4.302. 0.30 1.0 0.500 5.30 3. 0.31 0.7 0.490 8.90 4. 0.31 3.3 0.480 10, 8 5.0.40 0.5 0.425 21, 4 6. 0.40 3.5 0.367 31, 8

Table 2 has similar calculations as those in Table 1 but this time foran I-shaped TPM.

TABLE 2 (Σ_(max) = 0.556) a₁ ^(op) a₂ ^(op) Σ % 1. 0.15 2.0 0.538 6.302. 0.20 1.0 0.513 7.73 3. 0.31 0.7 0.471 14, 9 4. 0.31 3.3 0.412 25.9 5.0.40 0.5 0.415 25.4 6. 0.40 3.3 0.379 31.8

As can be seen from the above tables, it is established that theproposed algorithmic process, coupled with suitable computing means anduser selectable interfaces, it is possible to arrive at feasible ratiovalues that satisfy known constructive restrictions with optimum weightconfiguration.

Referring again back to Table 2, it is shown that the critical shapeefficiency factor and corresponding ratio values play a critical role inarriving at a TPM design that is weight optimized (as in items 1, 2 inTables 1 and 2), near weight optimized (items 3, 4 in Table 1 only), andnon-weight optimized (items 5, 6 in Table 1 and items 3-6 in Table 2).

It can be seen that the shape efficiency factor values of Σ₅ and of Σ₆based on the ratios in Variants 5 and 6, respectively, and which aresignificantly outside the ranges for the Z shape and the I shape TPMs,are significantly less than the theoretical maximum value Σ_(max) foreach shape.

It can be further seen from the shape efficiency factor values of Σ₁, Σ₂based on the ratios values in Variants 1, 2, respectively, and which arewithin the ranges for the Z shape and the I shape TPMs, that the shapeefficiency factor values are not significantly less than the theoreticalmaximum value Σ_(max) for a given shape.

It is to be understood that a TPM is implicitly associated with a shapeefficiency factor with theoretical maximum values. For various TPMshapes, these theoretical maximum values fall inside specified ranges.Inside the ranges, for each TPM shape, the values of the shapeefficiency factor have insignificant deviation from its theoreticalmaximum values and thus insignificant deviation from its theoreticalminimum weight values. Outside the ranges, the shape efficiency factorvalues decrease significantly, as a consequence of which so does theoverall weight in design. In all other regards, techniques for settingthe ratios themselves to feasible and workable dimensions and whichcomply with a given set of constructive restrictions is known and beyondthe scope of this invention.

In accordance with a further embodiment of the invention, various shapesof TPM can be compared in weight: the greater the maximum value of theshape efficiency factor Σ for a particular shape, the less is the TPMweight. For selecting the most efficient TPM shape with minimum weight,for each TPM of a given shape a plurality of maximum shape efficiencyfactors values Σ_(max)1, Σ_(max)2 . . . Σ_(max)N are determined, anoverall maximum shape efficiency factor value Σ_(0max) is determined andthe TPM of the shape having the overall maximum shape efficiency factorvalue Σ_(0max) is produced by known methods.

The proposed method of producing minimum weight transversal andlongitudinal reinforcing panels IV and V with respect to constructiverestrictions comprising the steps similar to method of producing aminimum weight TPM and including expressions (1)-(9).

1. TPM Analysis

1.1 Detailed Computational Analysis—TPM Selection

Analysis Scheme

For explanation purposes, an analysis scheme for a representative thinwall profile member has been selected that is very close to the existingefficient calculation method. For a TPM, such a method is the theory ofstability of compressed strips forming the TPM (with regard to theiroperation within supercritical range in respect of stability). Theselected analysis scheme predetermines the character and sequence ofdestruction of a compressed TPM.

A TPM is considered as a solid-walled structural member consisting ofplate (strips). Therefore it is possible that prior to the loss of thestability of the TPM as a whole (overall stability loss), the stripsforming the TPM would have suffered (local) buckling (which ischaracterized by warp of the cross-section, linearity and position ofcommon longitudinal ribs of the strips being maintained). This mayresult in destruction of the entire compressed TPM.

Geometrical irregularities (initial eccentricity of load application andbends) reduce critical stresses and, consequently, reduce theload-carrying capacity of the TPM. Calculations errors (of loads,stresses, deformations) and adopted assumptions of the selected analysisscheme are refined later in compliance with the Design Standards.

Optimization Model

What follows is a more detailed discussion of the operational flowdescribed in FIG. 16.

As already explained, the drawback common to all known calculationmethods of optimization is lack of a unified practical calculationmethod for multiple-parameter problems. In the thin wall systems optimumdesign theory, only those inverse problems of building mechanics havebeen examined that are the simplest in their mathematical descriptionwith only few simultaneously varied parameters.

To reduce the number of simultaneously varied parameters, the proposedmodel uses the following design approaches: 1) proceeding from simpledesign schemes to more complicated ones basing on analysis of variousforms of stability loss of the structure; 2) introduction ofgeneralizing coefficients (shape efficiency factor et al.); 3)employment of equal stability principle; 4) introduction of constructiverestrictions.

The suggested model using these design approaches enable analyticalrepresentation of the dimensions of cross-section (shape dimensions) ofcompressed TPM having the minimum weight.

The compressed TPM is considered as a “simple column” possessing hightorsional rigidity. As a rule, in the TPM-based structures their hightorsional rigidity is provided for. There are two kinds of stabilityloss—local and overall ones. Used to solve the problem is equatingdesign and critical stresses for both kinds of stability loss.Initially, applying the conditions of G. Cohen's theorem, it is provedthat the equations used ensure the minimum weight of a compressed TPM.

To equate the two kinds of stability loss as a criterion of the minimumweight, the following conditions formulated in the Cohen's theorem aresufficient:

The effective thickness function τ_(ef) (x_(i)) shall be independent onone of the cross-section dimensions x_(i), for example, x_(k); x_(i) areindependent dimensions determining the cross-section, where i=1, 2 . . .n.

Therefore,

$\frac{\partial{\overset{\_}{\tau}}_{ef}}{\partial{\overset{\_}{x}}_{k}} = 0$where${{\overset{\_}{\tau}}_{ef} = {\overset{\_}{t_{ef}}/\overset{\_}{q}}},{{{and}\mspace{14mu} \overset{\_}{t_{ef}}} = {t_{ef}/l_{re}}},{\overset{\_}{q} = {q/l_{re}}},$

t_(ef) being the effective thickness (area divided by unit length ofcross-section). Stresses σ_(cr. l)(x_(i)) and σ_(cr. tot)(x_(i)) shallbe monotony decreasing with x_(k).

Therefore,

${\frac{\partial{\sigma_{{cr}.l}\left( x_{i} \right)}}{\partial x_{k}} \neq 0};\mspace{14mu} {\frac{\partial{\sigma_{{cr}.{tot}}\left( x_{i} \right)}}{\partial x_{k}} \neq 0.}$

Let us check applicability of Cohen's equation for a compressed TPM. Forthis, thickness function τ_(ef) (x_(i)) and critical stress functionsσ_(cr. l)(x_(i)), σ_(cr. tot)(x_(i)) should be expressed in terms of TPMshape dimensions x_(i).

Let us introduce TPM dimensionless parameters a₁, a₂ . . . a_(n) asratios of TPM shape dimensions to characteristic dimensions.Characteristic dimensions a, δ_(a) shall be defined as the dimensions ofthe main strip appearing in the formula of local stability criticalstress.

FIG. 18 is a plot showing factor Ks versus ratios of U-shaped TPMdimensions.

FIG. 19 is another plot showing factor Ks versus ratios of U-shaped TPMdimensions.

FIG. 20 is yet another plot showing factor Ks versus ratios of U-shapedTPM dimensions.

FIG. 21 is a plot showing factor Ks versus ratios of Z-shaped TPMdimensions.

FIG. 22 is a plot showing factor Ks versus ratios of L-shaped TPMdimensions.

FIG. 23 is a plot showing factor Ks versus ratios of I-shaped TPM.

Let us express geometric characteristics of the profile in thedimensionless form through characteristic dimensions and dimensionlessparameters:

$\begin{matrix}{{{\overset{\_}{F}}_{s} = {\overset{\_}{a} \cdot {\overset{\_}{\delta}}_{a} \cdot {f_{1}\left( {a_{1},{a_{2}\mspace{14mu} \ldots}\mspace{14mu},a_{n}} \right)}}},} & \left( {1.1{.1}} \right) \\{{{\overset{\_}{J}}_{s} = {{\overset{\_}{a}}^{3} \cdot {\overset{\_}{\delta}}_{a} \cdot {f_{2}\left( {a_{1},{a_{2}\mspace{14mu} \ldots}\mspace{14mu},a_{n}} \right)}}},} & \left( {1.1{.2}} \right) \\{{{\overset{\_}{l}}_{s}^{2} = {{\overset{\_}{a}}^{2} \cdot {f_{3}\left( {a_{1},{a_{2}\mspace{14mu} \ldots}\mspace{14mu},a_{n}} \right)}}},} & \left( {1.1{.3}} \right) \\{{where}{{{{\overset{\_}{F}}_{s} = \frac{F_{s}}{l_{re}^{2}}};{{\overset{\_}{J}}_{s} = \frac{J_{s}}{l_{re}^{4}}};{\overset{\_}{i} = \frac{i_{s}}{i_{re}}};{\overset{\_}{a} = \frac{a}{l_{re}}};{{\overset{\_}{\delta}}_{a} = \frac{\delta_{a}}{l_{re}}}},}} & \;\end{matrix}$

f₁ (a₁, a₂, . . . a_(n)), f₂ (a₁, a₂, . . . a_(n)), f₃ (a₁, a₂, . . .a_(n)) are functions of dimensionless parameters of the TPM shape.

Then: the effective thickness

$\begin{matrix}{{{\overset{\_}{t}}_{ef} = {\frac{{\overset{\_}{F}}_{s}}{\overset{\_}{a}} = {{\overset{\_}{\delta}}_{a} \cdot {f_{1}\left( {a_{1},{a_{2}\mspace{14mu} \ldots}\mspace{14mu},a_{n}} \right)}}}},} & \left( {1.1{.4}} \right)\end{matrix}$

the effective thickness function:

$\begin{matrix}{{{\overset{\_}{\tau}}_{ef} = \frac{{\overset{\_}{\delta}}_{a} \cdot {f_{1}\left( {a_{1},{a_{2}\mspace{14mu} \ldots}\mspace{14mu},a_{n}} \right)}}{{\overset{\_}{q}}_{c}}},} & \left( {1.1{.5}} \right)\end{matrix}$

the local stability critical stress

$\begin{matrix}{{\sigma_{{cr}.l} = {\frac{K_{s} \cdot E}{\left( {\overset{\_}{a}/{\overset{\_}{\delta}}_{a}} \right)^{2}} \cdot \sqrt{\eta_{\delta}}}},} & \left( {1.1{.6}} \right)\end{matrix}$

K is the local stability factor of TPM shape:K_(s)=K_(s) (a₁, a₂ . . . , a_(n)), FIGS. 18-23:the overall stability critical stress

$\begin{matrix}{{\sigma_{{cr}.{tot}} = {{\frac{\pi^{2} \cdot E}{\lambda_{s}^{2}} \cdot \eta_{\sigma}} = {\pi^{2} \cdot E \cdot {\overset{\_}{a}}^{2} \cdot {f_{3}\left( {a_{1},{a_{2}\mspace{14mu} \ldots}\mspace{14mu},a_{n}} \right)} \cdot \eta_{\sigma}}}},} & \left( {1.1{.7}} \right)\end{matrix}$

where λ_(s) is TPM flexibility, λ_(s)=l_(re)/i

Assuming x_(k)=a and writing the partial derivatives as

$\frac{\partial{\overset{\_}{\tau}}_{ef}}{\partial\overset{\_}{a}},{\frac{\partial\sigma_{{cr}.l}}{\partial\overset{\_}{a}}\mspace{14mu} {and}\mspace{14mu} \frac{\partial\sigma_{{cr}.{tot}}}{\partial\overset{\_}{a}}},$

we see the Cohen's theorem conditions 1 and 2 are fulfilled. Therefore,equality of the critical stresses of local and overall stability for theTPM ensures its minimum weight.

1.2 Accounting for Material Properties at Elasticity Area (Derivation ofAnalytical Dependence for Elastic Ductility Factor)

To derive analytical formula for the elastic ductility (plasticity)factor for aluminum alloys, the following linear law of variation ofstress σ with flexibility λ_(s) is assumed:

$\begin{matrix}{{\sigma = {\sigma_{0} - {\frac{\sigma_{0} - \sigma_{pr}}{\lambda_{s}^{*}} \cdot \lambda_{s}}}},} & \left( {1.2{.1}} \right)\end{matrix}$

where λ_(s)* is the limiting flexibility of TPM,

$\lambda_{s}^{*} = {\pi \cdot {\sqrt{\frac{E}{\sigma_{pr}}}.}}$

On the other hand, TPM stress can be expressed in terms of flexibilityby equation (1.1.7). Excluding flexibility λ_(s) from equations (1.1.7)and (1.2.1), we obtain the ductility factor as follows:

$\begin{matrix}{\eta_{\sigma} = {\frac{\sigma}{\sigma_{pr}} \cdot {\left( \frac{\sigma_{0} - \sigma}{\sigma_{0} - \sigma_{pr}} \right)^{2}.}}} & \left( {1.2{.2}} \right)\end{matrix}$

It follows from (1.2.2) that the ductility factor η_(σ) depends onstress σ. To determine η_(σ) usually the successive approximation methodis used, which complicates the solution making the design process rathertime consuming. To avoid usage of this method, let us present criticalstress (1.1.6) and (1.1.7) in the following way:

σ_(cr.l)=σ_(cr.l(η) _(σ) ₌₁₎·√{square root over (η_(σ.l))}  (1.2.3),

σ_(cr.tot)=σ_(cr.tot(η) _(σ) ₌₁₎·η_(σ.tot)  (1.2.4),

where:σ_(cr.l(η) _(σ) ₌₁₎ is local stability critical stress (1.1.6)determined at ductility factor equal to 1;σ_(cr.tot(η) _(σ) ₌₁₎ is overall stability critical stress (1.1.7)determined at ductility factor equal to 1.

Solving (1.2.2) together with (1.2.3) and (1.2.4), respectively, andtaking into account equations σ_(cr.l)=σ_(cr.tot)=σ, we obtain:

$\begin{matrix}{{\eta_{\sigma.l} = \left\{ \frac{\left\lbrack {\sqrt{{\sigma_{pr} \cdot \left( {\sigma_{0} - \sigma_{pr}} \right)^{2}} + {4 \cdot \sigma_{{cr}.{l{({\eta_{\sigma} = 1})}}}^{2} \cdot \sigma_{0}}} - {\sigma_{pr}^{1/2} \cdot \left( {\sigma_{0} - \sigma_{pr}} \right)}} \right\rbrack^{2}}{4 \cdot \sigma_{{cr}.{l{({\eta_{\sigma} = 1})}}}^{3}} \right\}^{2}},} & \left( {1.2{.5}} \right) \\{\eta_{\sigma.{tot}} = {\frac{\sigma_{0}}{\sigma_{{cr}.{{tot}{({\eta_{\sigma} = 1})}}}} - {\frac{\sigma_{0} - \sigma_{pr}}{\sigma_{{cr}.{{tot}{({\eta_{\sigma} = 1})}}}} \cdot {\left( \frac{\sigma_{pr}}{\sigma_{{cr}.{{tot}{({\eta_{\sigma} = 1})}}}} \right)^{1/2}.}}}} & \left( {1.2{.6}} \right)\end{matrix}$

The sequence of determination of local and total critical stresses forTPM is as follows. First, the critical stress is determined at ductilityfactor value equaling 1 (one) σ_(cr.l(η) _(σ) ₌₁₎, σ_(cr.tot(η) _(σ)₌₁₎. If the obtained values of critical stress are less than σ_(pr), thecalculation is considered completed; if the value is greater, then usingequations (1.2.5) and (1.2.6) the elastic ductility factor isdetermined. After that, using equations (1.2.3) and (1.2.4) thesought-for value of critical stress is determined.

The similar sequence can be used to determine the analytical dependencefor elastic ductility factors of other materials. For example, for steelalloys these dependencies take on the form:

$\begin{matrix}{{\eta_{\sigma.l} = \frac{{\sigma_{0,2} \cdot \sigma_{{cr}.l.{({\eta_{\sigma} = 1})}}} - {\left( {\sigma_{0,2} - \sigma_{pr}} \right) \cdot \sigma_{pr}}}{\sigma_{{cr}.{l{({\eta_{\sigma} = 1})}}}^{2}}},} & \left( {1.2{.7}} \right) \\{\eta_{\sigma.{tot}} = {\frac{\sigma_{0,2} \cdot \sigma_{{cr}.{{tot}{({\eta_{\sigma} = 1})}}}}{{\left( {\sigma_{0,2} - \sigma_{pr}} \right) \cdot \sigma_{pr}} + \sigma_{{cr}.{{tot}{({\eta_{\sigma} = 1})}}}^{2}}.}} & \left( {1.2{.8}} \right)\end{matrix}$

1.3. Procedure for Designing Optimum Compressive TPM

We shall, in the present example, use equations for critical and designvalues of stress in order to determine the optimum characteristicdimensions ā, δ _(a).

σ_(c)=σ_(cr.l)=σ_(cr.tot)  (1.3.1),

where the design stress

$\begin{matrix}{{\sigma_{s} = {\frac{{\overset{\_}{P}}_{s}}{F_{s}} = \frac{{\overset{\_}{P}}_{s}}{\overset{\_}{a} \cdot {\overset{\_}{\delta}}_{a} \cdot {f_{1}\left( {a_{1},a_{2},{\ldots \mspace{14mu} a_{n}}} \right)}}}},{{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}}} & \left( {1.3{.2}} \right)\end{matrix}$

is the TPM stress factor.

Substituting (1.1.6), (1.1.7) and (1.3.2) into equations (1.3.1) andsolving them in respect to ā, δ _(a), we obtain:

$\begin{matrix}{{\overset{\_}{a} = {\left( \frac{{\overset{\_}{P}}_{s}}{\pi \cdot \sigma_{s}} \right)^{1/3} \cdot \frac{A_{1}}{\eta_{\sigma}^{1/12}}}},} & \left( {1.3{.3}} \right) \\{{{\overset{\_}{\delta}}_{a} = {\left( \frac{{\overset{\_}{P}}_{s}^{2} \cdot \pi}{\sigma_{s}^{2}} \right)^{1/3} \cdot \eta_{\sigma}^{1/12} \cdot A_{2}}},} & {\left( {1.3{.4}} \right),}\end{matrix}$

A₁, A₂ are functions of dimensionless parameters.

The functions of dimensionless parameters expressed through geometriccharacteristics of the shape and characteristic dimensions have thefollowing general form:

$\begin{matrix}{{A_{1} = \left( \frac{K_{s} \cdot {\overset{\_}{a}}^{4} \cdot {\overset{\_}{\delta}}_{a}^{2}}{{\overset{\_}{F}}_{s} \cdot {\overset{\_}{J}}_{s}} \right)^{1/6}},} & \left( {1.3{.5}} \right) \\{A_{2} = {\left( \frac{{\overset{\_}{a}}^{2} \cdot {\overset{\_}{\delta}}_{a}^{4} \cdot {\overset{\_}{J}}_{s}}{K_{s} \cdot {\overset{\_}{F}}_{s}^{5}} \right)^{1/6}.}} & \left( {1.3{.6}} \right)\end{matrix}$

Substituting equations (1.3.3) and (1.3.4) into (1.1.6) and taking intoaccount (1.3.2), we obtain the value of optimum stress

σ_(s)=(E ³·σ⁴ ·P _(s) ²)^(1/5)·√{square root over (η_(σ))}·A  (1.3.7),

A is the shape efficiency factor, which is the function of dimensionlessparameters a₁, a₂ . . . , a_(n).

Here, the ductility factor η_(σ) is obtained from the formula (1.2.5) byreplacing σ_(cr.l.(η) _(σ) ₌₁₎ with σ_(s(η) _(σ) ₌₁₎.

Expressing A through geometrical characteristics of TPM andcharacteristic dimensions, we obtain:

$\begin{matrix}{A = {\left\lbrack {\left( \frac{{\overset{\_}{J}}_{s}}{{\overset{\_}{F}}_{s}} \right)^{2} \cdot \frac{K}{\left( {\overset{\_}{a}/{\overset{\_}{\delta}}_{a}} \right)^{2}}} \right\rbrack^{1/5}.}} & \left( {1.3{.8}} \right)\end{matrix}$

The dimensionless weight of TPM can be expressed as:

$\begin{matrix}{{{{\overset{\_}{G}}_{s} = \frac{{\overset{\_}{P}}_{s}}{\sigma_{s}}},{where}}{{\overset{\_}{G}}_{s} = {\frac{G_{s}}{l_{re}^{2} \cdot l_{s} \cdot \gamma}.}}} & {\left( {1.3{.9}} \right),}\end{matrix}$

As one can see from (1.3.9), the minimum weight is obtained at themaximum stress in TPM. It follows from (1.3.7) that the only way toincrease σ_(s) is to increase the shape efficiency factor A.

Let us consider the physical meaning of the shape efficiency factor. Theratio

$f_{s} = \frac{J_{s}}{F_{s}^{2}}$

is called the TPM “shape factor”. The more is the material “spaced”(that is, the higher is its distance from the neutral axis), the higheris its shape factor f_(s). On the other hand, however, the value

$\frac{K_{s}}{\left( {a/\delta_{a}} \right)^{2}}$

characterizing the TPM local stability is thereby reduced. As a result,the shape efficiency factor possesses a maximum.

Specifying a series of values a₁, a₂ . . . , a_(n), it is possible todetermine the maximum value A_(max) and optimum (in weight) values ofdimensionless parameters a₁ ^(op), a₂ ^(op) . . . , a_(n) ^(op)corresponding to it. Substituting the maximum values of the shapeefficiency factor A_(max) into equation (1.3.7), we obtain the maximumstress, and using (1.3.9), the minimum weight.

Then, substituting the optimum values of dimensionless parameters a₁^(op), a₂ ^(op) . . . , a_(n) ^(op) and values K_(s) ^(op)=K_(s) ^(op)(a₁ ^(op), a₂ ^(op) . . . , a_(n) ^(op)) corresponding to these intoequations (1.3.3) and (1.3.4), we can find the optimum characteristicdimensions a^(op), δ_(a) ^(op). The other of TPM shape dimensions areessentially functions of characteristic dimensions a^(op), δ_(a) ^(op)and dimensionless parameters a₁ ^(op), a₂ ^(op) . . . , a_(n) ^(op).

Slight deviations of A with respect to A_(max) (i.e. small deviations ofweight with respect to G_(min)) could entail considerable difference inTPM shape absolute dimensions. This important conclusion can be employedto meet constructive requirements as to the dimensions of the shapeprovided maximum material saving.

1.4 Determination of Optimum Parameters of Compressed TPM of the MostCommon Shapes

In this section, we describe determination of parameters of U-, Z-,L-shape TPMs for two aluminum alloys, D16-T and B-95 (Table 1.1 in FIG.41), and some parameters I-shaped TPM for steel.

FIG. 41 is a table showing feasible dimensions of U-shaped TPM.

The results are presented in the form of plots of optimum parametersversus stress factor, which enables to easily find the optimum actualcross-section dimensions and minimum weight of TPM.

FIG. 24 is a plot showing shape efficiency factor A versus ratios ofU-shaped TPM dimensions.

FIG. 25 is plot showing shape efficiency factor A versus ratios ofU-shaped TPM dimensions.

FIG. 26 is yet plot showing shape efficiency factor A versus ratios ofU-shaped TPM dimensions.

FIG. 27 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{a} = \frac{a}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

FIG. 28 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{\delta_{a}} = \frac{\delta_{a}}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

FIG. 29 is a plot showing optimum stress σ_(s) versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

FIG. 30 is a plot showing dimensionless weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

U-shaped TPM.

To meet the constructive requirements provided small variations of shapeefficiency factor A (and hence, of TPM weight) as compared with itsmaximum value A_(max), the TPM shape dimensions could be varied.

Feasible dimensions for U-shaped TPM taking into account certainconstructive requirements are presented in Table 1.1 (FIG. 41).

For TPM stress factors used in aviation P _(s)=(0.2-1) daN/cm², at anoptimum value of

${a_{1}^{op} = {\frac{c}{b} = 0.2}},$

the length of a flange c can be insufficient to mount a rivet (c<2 cm),so a feasible value of a₁ ^(op)=0.3 is selected. Provided optimumrelationship between thickness values

${a_{3}^{op} = {\frac{1}{a_{4}^{op}} = {\frac{\delta_{c}}{\delta_{b}} = {\frac{\delta_{a}}{\delta_{b}} = 2}}}},$

thickness δ_(b) can prove too low value (less than 1 mm), which makes itdifficult to manufacture such a profile in mass-scale production. Themost suitable for production is a constant thickness profile with

${a_{3} = {\frac{1}{a_{4}^{op}} = {\frac{\delta_{c}}{\delta_{b}} = {\frac{\delta_{a}}{\delta_{b}} = 1}}}},$

(profile No. 1, Table 1.1), but at some stress factors value the flangethickness value δ_(c) may be less of the skin thickness value, which isnot recommended. Therefore, profile No. 2 is selected as the mostfeasible one.

At low values of the TPM stress factor, the thickness of the U-shapedTPM webs can to be very low value and technologically not feasible. Ifthis is the case, Z-shaped and L-shaped profiles with the configurationopen with the skin sheet should be selected.

Let us consider Z-shaped TPM with four variable dimensions b, δ_(b), c,δ_(c).

FIG. 31 is a plot showing shape efficiency factor A versus ratios ofZ-shaped TPM dimensions.

The plot of shape efficiency factor A for various values of a₁, a₂ arepresented in FIG. 31.

Respecting constructive reasons, for example, so that a rivet could fit,it is assumed that a₁ ^(op)=0.3. With this, the maximum valueA_(max)=0.515; a₂ ^(op)=2. Then the factor K_(s) ^(op)=5.8 (FIG. 31).

The calculation results are presented in plots shown in FIGS. 32 to 35.

FIG. 32 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{b} = \frac{b}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 33 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{\delta_{b}} = \frac{\delta_{b}}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 34 is a plot showing optimum stress σ_(s) versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 35 is a plot showing dimensionless weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

Z-shaped TPM.

FIG. 36 is a plot showing shape efficiency factor A versus ratio ofL-shaped TPM dimensions.

The plot for the shape efficiency factor for L-shaped TPM is presentedin FIG. 36. Basing on design reasons, namely, rivets mountingconditions, it is assumed that: a₁ ^(op)=0.3. With this, the maximumvalue A_(max)=0.328; a₂ ^(op)=1. Then the factor K_(s) ^(op)=0.55 (FIG.22).

Further, calculated are the factors A₁ ^(op)=1.2; A₂ ^(op)=0.641. Usingequations (1.3.7), (1.3.3), (1.3.4) and (1.3.9), we can determine theoptimum stress, characteristic dimensions and minimum weight.

FIG. 37 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{b} = \frac{b}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

FIG. 38 is a plot showing scaled (dimensionless) characteristicdimension

$\overset{\_}{\delta_{b}} = \frac{\delta_{b}}{l_{re}}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

FIG. 39 is a plot showing optimum stress σ_(s) versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

FIG. 40 is a plot showing dimensionless weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus TPM stress factor

${{\overset{\_}{P}}_{s} = \frac{P_{s}}{l_{re}^{2}}},$

L-shaped TPM.

The similar procedure is used to determine the optimum parameters of TPMof I shape.

The values of the factor K_(s) are presented in FIG. 23.

2. TPM-Panel Analysis

Detailed Computational Analysis: TPM-Panel Selection

2.1 Analyses Scheme—Stringer Panel with Buckling Skin

For explanation purposes, an analysis scheme for a representativestringer panel with buckling skin is presented.

Panel design comes down to finding of six dimension of its cross-sectionb, δ_(b), c, δ_(c), b_(s), δ given the linear compressive force q_(p),overall dimensions B, l and panel material.

FIG. 42 shows a panel with Z-shaped TPM.

Selection of the panel cross-section dimensions allows for infinitenumber of solution variants. Out of these, the one is selected meetingthe minimum weight panel condition.

TPM shape dimensions b, δ_(b), c, δ_(c), are determined according toequations (1.3.3), (1.3.4), given the force P_(s) per one stringer. Theunknown variables in this case would be pitch of TPM installation b_(s),skin thickness δ and force P_(s) per one stringer.

The panel weight G_(s.p) is determined by the sum of weight of TPM(stringers) G_(s) and the weight of the skin G_(sk) according to theequation:

$\begin{matrix}{G_{s.p.} = {{G_{s} + G_{sk}} = {\left( {\frac{P_{s}}{\sigma_{s} \cdot b_{s}} + \delta} \right) \cdot l_{s} \cdot B \cdot {\gamma.}}}} & \left( {2.1{.1}} \right)\end{matrix}$

Let us express the panel weight (2.1.1) in the dimensionless form:

$\begin{matrix}{{{\overset{\_}{G}}_{s.p.} = {\frac{{\overset{\_}{P}}_{s}}{\sigma_{s} \cdot {\overset{\_}{b}}_{s}} + \overset{\_}{\delta}}}{{{{where}\mspace{14mu} {\overset{\_}{G}}_{s.p.}} = \frac{G_{s.p.}}{l_{s} \cdot l_{re} \cdot B \cdot \gamma}},{\overset{\_}{\delta} = \frac{\delta}{l_{re}}},{{\overset{\_}{b}}_{s} = {\frac{b_{s}}{l_{re}}.}},}} & \left( {2.1{.2}} \right)\end{matrix}$

The force per one stringer is related to the panel parameters by theexpression:

P _(s) =q _(p) ·b _(s) −δ·b _(s)·σ_(av)  (2.1.3),

σ_(av) being the average stress in the skin between TPM.

Rewriting (2.1.3) in the dimensionless form, we obtain the TPM stressfactor as:

P _(s) =b _(s)·( q _(p)−δ·σ_(av))  2.1.4),

q _(p) is the panel stress factor:

${\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}$

The value σ_(av) is determined according to the formula:

$\begin{matrix}{{\sigma_{av} = \sqrt{\sigma_{s} \cdot \sigma_{{cr}.{sk}}}},{where}} & \left( {2.1{.5}} \right) \\{{\sigma_{{cr}.{sk}.} = \frac{K_{p} \cdot E}{\left( {{\overset{\_}{b}}_{s}/\overset{\_}{\delta}} \right)^{2}}},} & \left( {2.1{.6}} \right)\end{matrix}$

Here, the factor K_(p)=3.6 for pivoted leaning of the skin.

Let us eliminate from the expression (2.1.4) the average skin stressσ_(av). For this purpose, substitute (2.1.5) into (2.1.4), taking intoconsideration (2.1.6); we get

P _(s) =q _(p) ·b _(s)−δ ²·√{square root over (K _(p) ·E·σ_(s))}  (2.1.7).

The stress in the compressed isolated TPM σ_(s) is related to the stressfactor P _(s) by the expression (1.3.7). Equation (1.3.7) enables toconsider as a sought-for parameter the stringer stress σ_(s) rather thanthe stress factor P _(s).

Substituting (2.1.7) into (1.3.7), we get the constraint equation forthe three sought-for parameters σ_(s), b _(s), δ:

σ_(s) =[E ³·π⁴·( q _(p) ·b _(s)−δ ²·√{square root over (K _(p) ·E·σ_(s))})²]^(1/5)·√{square root over (η_(σ))}·A  (2.1.8).

The obtained equation can be rewritten in the form (1.2.3):

σ_(s)=σ_(s(η) _(σ) ₌₁₎·√{square root over (η_(σ))},

then the ductility factor η_(σ) can be obtained from the equation(1.2.5), replacing therein η_(σ.l) with η_(σ) and σ_(cr.l(η) _(σ) ₌₁₎with σ_(s(η) _(σ) ₌₁₎.

Equation (2.1.8) cannot be solved with respect to σ_(s). Given thevalues b _(S) and δ from (2.1.8), one can numerically determine thevalue of σ_(s) and then calculate P _(s) through (1.3.7). Substitutingvalues of σ_(s) and P _(s) into (2.1.1), we obtain the panel weight.

Numerical solution of (2.1.8) with respect to σ_(s) is rathercomplicated and does not enable to obtain an analytical expression forthe weight. The analytical expression for the panel weight as a functionof two parameters σ_(s) and δ could be obtained as follows.

Solving (2.1.8) with respect to b _(s), we get:

$\begin{matrix}{{\overset{\_}{b}}_{s} = {\frac{1}{{\overset{\_}{q}}_{p}} \cdot {\left\lbrack {\sqrt{\left( \frac{\sigma_{s}}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{5} \cdot \frac{1}{E^{3} \cdot \pi^{4}}} + {{\overset{\_}{\delta}}^{2} \cdot \sqrt{K_{p} \cdot E \cdot \sigma_{s}}}} \right\rbrack.}}} & \left( {2.1{.9}} \right)\end{matrix}$

From (1.3.7) we get the TPM stress factor:

$\begin{matrix}{{\overset{\_}{P}}_{s} = {\sqrt{\left( \frac{\sigma_{s}}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{5} \cdot \frac{1}{E^{3} \cdot \pi^{4}}}.}} & \left( {2.1{.10}} \right)\end{matrix}$

Substituting (2.1.9) and (2.1.10) into (2.1.2), we get the analyticalexpression for the dimensionless panel weight as a function of σ_(s) andδ in the following form:

$\begin{matrix}{{\overset{\_}{G}}_{s.p.} = {\frac{{\overset{\_}{q}}_{p} \cdot \sigma_{s}}{{\left( {\pi \cdot E} \right)^{2} \cdot {\overset{\_}{\delta}}^{2} \cdot K_{p}^{\frac{1}{2}} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}}} + \sigma_{s}^{2}} + {\overset{\_}{\delta}.}}} & \left( {2.1{.11}} \right)\end{matrix}$

The ductility factor η_(σ) in (2.1.11) for σ_(s)>σ_(pr) is determinedfrom the equation (1.2.5) replacing therein σ_(cr.l(η) _(σ) ₌₁₎ withσ_(s(η) _(σ) ₌₁₎.

FIG. 43 is a graph showing dimensionless panel weight

${\overset{\_}{G}}_{s} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus TPM stress σ_(s), U-shaped TPM.

Some of the results of calculations according to (2.1.11) are presentedin FIG. 43.

The TPM shape efficiency factor used, A is that for the Z-shapedprofile, A=0.515. The value of the shape efficiency factor variesinsignificantly with changing of the TPM shape and does not affectconsiderably the results of calculations according to (2.1.11).

As these calculations demonstrate, with the dimensionless skin thicknessvalues typical for aircraft industry, δ=0.001-0.005, the minimum panelweight is obtained for the minimum TPM stress σ_(s).

Proceeding from the stress to the design panel cross-section dimensions,we see that with reducing σ_(s), the distance between TPMs (pitch ofstringer longitudinal installation) (2.1.9) is also reduced, and thedimensions of the TPM shape are also reduced, due to reduction of theTPM shape factor (2.1.10). Therefore, to reduce the panel weight oneshould seek to increase the number of TPMs (reduction of the pitch ofstringer longitudinal installation) and reduce the TPM shapecross-section dimensions.

The lower limit value for the pitch of longitudinal installation of TPMis determined by the value of the critical skin stress σ_(cr.sk)(2.1.6). The critical stress in the skin cannot exceed the criticalstress in the TPM. Therefore, the optimum panel weight is achievedprovided equality of critical stresses of skin and TPM, that is, rulingout skin buckling.

The lower limit value for reducing TPM shape dimensions is determined bythe minimum TPM shape moment of inertia, providing for the wavelength(in stability loss) in the panel width equal to or less than thedistance between the rivet joint of TPM fastening to the skin. Reductionof TPM shape dimensions is evidently restricted also by the constructiverestrictions to TPM.

2.2 Design Procedure for Stringer

Panel of Theoretical Minimum Weight

It has been proved in 2.1 that the stringer panel reaches the minimum inweight provided equality of critical stress for skin and stringer, viz.if:

σ_(cr.sk)=σ_(s)=σ  (2.2.1).

Taking (2.2.1) into account, the dimensionless width of the skin betweenthe rivet joints of TPM fastening (for the open-configuration TPM equalto its dimensionless pitch of longitudinal installation) shall bewritten as (2.1.6):

$\begin{matrix}{{\overset{\_}{b}}_{sk} = {{\overset{\_}{b}}_{s} = {\overset{\_}{\delta} \cdot {\sqrt{\frac{K_{p} \cdot E}{\sigma}}.}}}} & \left( {2.2{.2}} \right)\end{matrix}$

The dimensions of the panel cross-section have been shown in 2.1 to befunctions of three parameters σ_(s), b _(s) and δ. Let us determinethese parameters using the derived equations (2.2.1) and (2.2.2).

The TPM stress factor (2.1.4) taking into account (2.2.1) shall bewritten as:

P _(s) =b _(s)·( q _(p)−δ·σ)  (2.2.3).

Taking into consideration (2.2.3), the expression for the panel weighttakes on the form:

$\begin{matrix}{{\overset{\_}{G}}_{s.p.} = {\frac{{\overset{\_}{q}}_{p}}{\sigma}.}} & \left( {2.2{.4}} \right)\end{matrix}$

Expression (2.2.4) implies that for a panel with the non-buckling skin,the minimum weight is reached at the maximum panel stress.

For the maximum stress, the yield stress for the panel material isadopted:

σ=σ_(0.2)  (2.2.5).

Henceforth, one of the sought-for parameters, the panel stress σ, hasbeen determined.

Given the panel stress σ, one can determine the TPM shape dimensions. Tothis end, we shall find the TPM stress factor (2.1.10) taking intoaccount (2.2.5) and (2.2.1):

$\begin{matrix}{{\overset{\_}{P}}_{s} = {\sqrt{\left( \frac{\sigma_{0,2}}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{5} \cdot \frac{1}{E^{3} \cdot \pi^{4}}}.}} & \left( {2.2{.6}} \right)\end{matrix}$

It follows from (2.2.6) that given the material and the shape efficiencyfactor A, the TPM stress factor P _(s) is a constant.

The optimum characteristic dimensions of the shape, taking into account(2.2.6), shall be determined from equations (1.3.3) and (1.3.4). Thecharacteristic dimensions of the TPM shape are constant values, whileabsolute dimensions depend linearly from the panel length(1.1.1)-(1.1.3).

The second sought-for parameter, the dimensionless pitch of longitudinalinstallation b _(s), can be determined from the equation (2.2.2), shouldthe third sought-for parameter be found, namely, the dimensionless skinthickness δ. Expressions (2.2.1) and (2.2.5) enable finding thedimensionless skin thickness.

Substituting (2.2.2) into the constraint equation (2.1.8) and takinginto consideration (2.2.5), we get:

$\begin{matrix}{{\left( \frac{\sigma_{0,2}}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot \frac{1}{E^{\frac{3}{2}} \cdot \pi^{2}}} = {\overset{\_}{\delta} \cdot \left( \frac{K_{p} \cdot E}{\sigma_{0,2}} \right)^{\frac{1}{2}} \cdot {\left( {{\overset{\_}{q}}_{p} - {\overset{\_}{\delta} \cdot \sigma_{0,2}}} \right).}}} & \left( {2.2{.7}} \right)\end{matrix}$

Solution of (2.2.7) with respect to δ leads to the square equation ofthe kind:

$\begin{matrix}{{{{{\lambda \cdot {\overset{\_}{\delta}}^{2}} - {B \cdot \overset{\_}{\delta}} + \gamma} = 0}{{where}\text{:}}\lambda = \left( {K_{p} \cdot E \cdot \sigma_{0,2}} \right)^{\frac{1}{2}}},{\beta = {{\overset{\_}{q}}_{p} \cdot \left( \frac{K_{p} \cdot E}{\sigma_{0,2}} \right)^{\frac{1}{2}}}},{\gamma = {\left( \frac{\sigma_{0,2}}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot {\frac{1}{E^{\frac{3}{2}} \cdot \pi^{2}}.}}}} & \left( {2.2{.8}} \right)\end{matrix}$

Solution of (2.2.8) is

$\begin{matrix}{{\overset{\_}{\delta}}_{1,2} = {\frac{\beta}{2 \cdot \lambda} \pm {\left( {\frac{\beta}{4 \cdot \lambda^{2}} - \frac{\gamma}{\lambda}} \right)^{\frac{1}{2}}.}}} & \left( {2.2{.9}} \right)\end{matrix}$

Substituting into (2.2.9) the values of λ, β, γ, we get:

$\begin{matrix}{{\overset{\_}{\delta}}_{1,2} = {\frac{{\overset{\_}{q}}_{p}}{2 \cdot \sigma_{0,2}} \pm {\sqrt{\frac{{\overset{\_}{q}}_{p}^{2}}{4 \cdot \sigma_{0,2}^{2}} - {\left( \frac{\sigma_{0,2}}{E \cdot \pi} \right)^{2} \cdot \frac{1}{\left( {A \cdot \sqrt{\eta_{\sigma}}} \right)^{\frac{5}{2}} \cdot K_{p}^{\frac{1}{2}}}}}.}}} & \left( {2.2{.10}} \right)\end{matrix}$

It follows from (2.2.10) that for the same load there exist two valuesof the dimensionless skin thickness corresponding to the panel minimumweight.

FIG. 45 is a stringer panel with U-shaped TPM.

For the U-shaped (forming with the sheet a closed configuration) TPM,the dimensionless pitch of longitudinal installation of TPM isdetermined from the following equation:

$\begin{matrix}{{\overset{\_}{b}}_{s} = {{{\overset{\_}{b}}_{re} + \overset{\_}{a} + \overset{\_}{c}} = {{\overset{\_}{\delta} \cdot \left( \frac{K_{p} \cdot E}{\sigma} \right)^{\frac{1}{2}}} + \overset{\_}{a} + {\overset{\_}{c}.}}}} & \left( {2.2{.11}} \right)\end{matrix}$

Then the constraint equation (2.1.8) for the closed profile shall takeon the form:

$\begin{matrix}{{\left( \frac{\sigma}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot \frac{1}{E^{\frac{3}{2}} \cdot \pi^{2}}} = {\left\lbrack {{\overset{\_}{\delta} \cdot \left( \frac{K_{p} \cdot E}{\sigma} \right)^{\frac{1}{2}}} + \overset{\_}{a} + \overset{\_}{c}} \right\rbrack \cdot {\left( {{\overset{\_}{q}}_{p} - {\overset{\_}{\delta} \cdot \sigma}} \right).}}} & \left( {2.2{.12}} \right)\end{matrix}$

The formula to determine the dimensionless thickness of the skin for TPMforming with the sheet a closed configuration shall take on the form:

$\begin{matrix}{\delta_{1,2} = {\frac{{\overset{\_}{q}}_{p}}{2 \cdot \sigma_{0,2}} - {{\frac{\overset{\_}{a} + \overset{\_}{c}}{2} \cdot \left( \frac{\sigma_{0,2}}{K_{p} \cdot E} \right)^{\frac{1}{2}}} \pm {\pm {\sqrt{\begin{matrix}{\left\lbrack {\frac{{\overset{\_}{q}}_{p}}{2 \cdot \sigma_{0,2}} - {\frac{\overset{\_}{a} + \overset{\_}{c}}{2} \cdot \left( \frac{\sigma_{0,2}}{K_{p} \cdot E} \right)^{\frac{1}{2}}}} \right\rbrack^{2} -} \\{\frac{1}{E^{2} \cdot \pi^{2} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot K_{p}^{\frac{1}{2}}} + \frac{{\overset{\_}{q}}_{p} \cdot \left( {\overset{\_}{a} + \overset{\_}{c}} \right)}{\left( {K_{p} \cdot E \cdot \sigma_{0,2}} \right)^{\frac{1}{2}}}}\end{matrix}}.}}}}} & \left( {2.2{.13}} \right)\end{matrix}$

The analysis of the radicands in (2.2.10) and (2.2.13) gives the rangeof admissible values of the panel stress for which optimum values forthe skin thickness can be obtained.

FIG. 57 are tables showing (i) a range of allowable values for stringerpanel stress factors, and (ii) features of real and optimum-designpanels.

It follows from the Table 2.1 shown in FIG. 57 that for TPM forming withthe sheet a closed configuration the range of admissible values for thepanel stress factors is narrower than that for the open profiles.Actually, the feasible range for the panel stress factors shall be evenless, as the optimum dimensionless values for the skin thicknessobtained from equations (2.2.10) and (2.2.13) could prove so small thattheir manufacturing would be impractical.

The panel stress factors pertinent to the most part of aircraft designsconstitute q_(p)=(20-60) daN/cm². It follows from Table 2.1 thatfeasible stress factor for a minimum weight panel exceeds the abovevalues. In the following section it will be shown how to find theoptimum panel parameters for any feasible stress factor values.

2.3 Design Procedure for Stringer Panel of Feasible Minimum Weight

If a stringer panel shall be designed for any feasible linearcompressive loads q_(p), or if dimensions of the panel cross-section donot meet the constructive restrictions, then the optimum stress shall beconsidered the stress σ_(n), which would be less than the yield stress,that is, σ_(n)<σ_(0.2). In this case the panel would possess the leastfeasible weight rather than theoretical minimum weight (i.e. the minimumweight accounting for said restrictions). The value of the stress σ_(n)should be as close to σ_(0.2) as possible (2.2.4), with this, the TPMstress factor P _(s) would approach its maximum value (2.2.6). Thismaximum value of the TPM stress factor P _(s) provided panel stressfactor q _(p) being constant and panel stress σ being at its maximum isonly possible with the minimum dimensionless skin sheet thickness δ=δ_(min) (2.2.3).

Let us determine the stress for the least feasible weight panel. To thisend, in the equation (2.2.7) the equations σ_(0.2)=σ_(n) and δ=δ _(min)shall be accounted for; for the TPM forming with the sheet an openconfiguration we get:

$\begin{matrix}{{{\left( \frac{\sigma_{n}}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot \frac{1}{E^{\frac{3}{2}} \cdot \pi^{2}}} = {{\overset{\_}{\delta}}_{\min} \cdot \left( \frac{K_{p} \cdot E}{\sigma_{n}} \right)^{\frac{1}{2}} \cdot \left( {{\overset{\_}{q}}_{p} - {{\overset{\_}{\delta}}_{\min} \cdot \sigma_{n}}} \right)}},} & \left( {2.3{.1}} \right)\end{matrix}$

σ_(n) being the maximum feasible panel stress provided minimum skinthickness δ_(min).

Solving (2.3.1) with respect to σ_(n), we get the cubic equation of thetype

$\begin{matrix}{{{{\sigma_{n}^{3} + {3 \cdot U \cdot \sigma_{n}} + {2 \cdot V}} = 0},{where}}{{U = {\frac{1}{3} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot K_{p}^{\frac{1}{2}} \cdot \left( {E \cdot \pi} \right)^{2} \cdot {\overset{\_}{\delta}}_{\min}^{2}}},{V = {{- \frac{1}{2}} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot K_{p}^{\frac{1}{2}} \cdot \left( {E \cdot \pi} \right)^{2} \cdot {\overset{\_}{\delta}}_{\min} \cdot {{\overset{\_}{q}}_{p}.}}}}} & \left( {2.3{.2}} \right)\end{matrix}$

The discriminant of (2.3.2) is determined from the equation:

D=V ² +U ³  (2.3.3).

The discriminant (2.3.3) is positive; therefore equation (2.3.2) shallhave the single real root determined from Cardan formula:

σ_(n)=β+γ  (2.3.4),

where β=(√{square root over (V²+U³)}−V)^(1/3), γ=(−√{square root over(V²+U³)}−V)^(1/3).

The ductility factor η_(σ) appearing in the equation for the stress(2.3.4) shall be determined from equation (1.2.2), replacing therein σwith σ_(n); we get

$\begin{matrix}{\eta_{\sigma} = {\frac{\sigma_{n}}{\sigma_{pr}} \cdot {\left( \frac{\sigma_{0} - \sigma_{n}}{\sigma_{0} - \sigma_{pr}} \right)^{2}.}}} & \left( {2.3{.5}} \right)\end{matrix}$

One can deduce from (2.3.5) that the ductility factor η_(σ) appearing inthe formula for the sought-for stress σ_(n) (2.3.4), in its turn, isdependent from the stress σ_(n). Therefore, σ_(n) and η_(σ) are soughtfor by the successive approximation method in the following way.

In the first approximation, we assume η_(σl)=1, and through the equation(2.3.4) we determine the first approximation of the TPM stress σ_(nl).If σ_(nl)<σ_(pr), then given σ_(nl), from (2.3.5) we get the secondapproximation for the ductility factor η_(σ2), and so on. Thecalculation is terminated provided successive approximations for stressσ_(n) (or η_(σ)) would differ by a preset value.

For the TPM forming with the sheet a closed configuration, theexpression (2.2.12) takes on the form:

$\begin{matrix}{{{\left( \frac{\sigma_{n}}{\sqrt{\eta_{\sigma}} \cdot A} \right)^{\frac{5}{2}} \cdot \frac{1}{E^{\frac{3}{2}} \cdot \pi^{2}}} = {2 \cdot {\overset{\_}{\delta}}_{\min} \cdot \left( \frac{K_{p} \cdot E}{\sigma_{n}} \right)^{\frac{1}{2}} \cdot \left( {{\overset{\_}{q}}_{p} - {{\overset{\_}{\delta}}_{\min} \cdot \sigma_{n}}} \right)}},} & \left( {2.3{.6}} \right)\end{matrix}$

it is assumed here that

b _(s)=2· b _(re)  (2.3.7).

The values of coefficients U and V of the cubic equation (2.3.2) for theU-shaped TPM will be twice higher. The maximum feasible panel stress forthe minimum skin thickness δ _(min) shall be determined, similarly tothat of the open profiles, from equation (2.3.4). The equation (2.3.4)enables finding the panel stress given the minimum dimensionless skinthickness, hence, determining all the other parameters of the panelcross-section.

The minimum dimensionless skin thickness is determined from theexpression

${\overset{\_}{\delta}}_{\min} = {\frac{\delta_{\min}}{l_{re}}.}$

Here, l_(re)=l_(s) (for the factor accounting for stringer leaning onrib c=1), while the skin thickness is assumed δ_(min)=(0.8-1) mm. Forsmaller skin thickness the countersunk-headed rivets are not employed,as they do not ensure adequate skin fastening strength. Cup-headedrivets are not employed due to aerodynamic restrictions.

2.4 Calculation of Optimum Parameters for a Stringer Panel

Considered here are algorithms for design of stringer panel of optimumweight. Some of the calculation results are presented as plots ofoptimum parameters versus the panel stress factor. The plots arepresented for U-shaped, Z-shaped and L-shaped profiles.

Algorithm of Design for the Theoretical Minimum Weight Stringer Panel

1. An array of panel stress factor values q _(p) is preset.2. The ductility factor is calculated through equation (1.2.2) replacingtherein σ with σ_(0.2).3. TPM stress factor is calculated through equation (2.2.6).4. Optimum characteristic scaled (dimensionless) shape dimensions forTPM are calculated through equations (1.3.3) and (1.3.4) (withσ_(s)=σ_(0.2)). The absolute shape dimensions are obtained multiplyingthe scaled dimension values by l_(re) (1.1.3).5. From equations (2.2.10) and (2.2.13), the optimum dimensionless skinthickness is calculated.6. From equations (2.2.2) and (2.2.11), the optimum dimensionless pitchof stringer installation is calculated (with σ=σ_(0.2)).7. From equation (2.2.4), the optimum dimensionless minimum weight ofthe stringer panel is calculated (with σ=σ_(0.2)).

FIG. 44 is a graph showing dimensionless skin thickness

$\overset{\_}{\delta} = \frac{\delta}{l_{re}}$

versus the panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

FIG. 46 is a graph showing dimensionless distance between stringers

$\overset{\_}{b_{s}} = \frac{b_{s}}{l_{re}}$

versus the panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

The plots of optimum value of the skin thickness δ and pitch of stringerinstallation b _(s) are presented in FIGS. 44 and 46. Out of the twovalues of the optimum dimensionless skin thickness δ and pitch ofstringer installation b _(s) given a particular value of the panelstress factor, those values of δ and b _(s) are selected that meet theconstructive requirements the best.

FIG. 47 is a plot showing TPM shape width a, b, c versus the panelstress factor

${{\overset{\_}{q}}_{p} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

FIG. 48 is a plot showing TPM shape thickness δ_(a), δ_(b), δ_(c) versusthe panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum weight panel).

The plots of optimum absolute dimensions of profiles a, b, c, δ_(a),δ_(b), δ_(c) are presented in FIGS. 47, 48.

FIG. 49 is a plot showing dimensionless panel weight

$\overset{\_}{G_{s}} = \frac{G_{s}}{l_{s} \cdot l_{re}^{2} \cdot \gamma}$

versus the panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}}},$

materials D16-T and B95 (least feasible weight panel).

The plot of the dimensionless minimum weight of the stringer panel G_(s.p.) with TPM of any shape is presented in FIG. 49.

Algorithm of Design for the Minimum Feasible Weight Stringer Panel

1. An array of panel stress factor values q_(p) is preset.2. The minimum dimensionless skin thickness is calculated:

δ _(min)=δ_(min) /l _(re)

Assumed for the calculation are: l_(re)=1 m, δ_(min)=1 mm, minimumdimensionless skin thickness δ _(min)=0.001.3. From equation (2.3.4), stress σ_(n) is determined (if σ_(n)>σ_(pr),then σ_(n) is determined through the consecutive approximations methodthrough formula (2.3.5)).4. From equation (2.1.10) the stress factor Ps for TPM is calculated(here, σ_(s)=σ_(n)).5. From equations (1.3.3) and (1.3.4), the optimum characteristic scaledTPM shape dimensions are calculated (here, σ_(s)=σ_(n)).6. The optimum dimensionless distance (pitch) between stringers iscalculated through equations (2.2.2) and (2.2.11) (here, σ=σ_(n)).7. The least panel weight is calculated from equation (2.2.4) (here,σ=σ_(n)). Some of the calculation results are presented in plots, FIGS.50 to 54.

FIG. 50 is a plot showing dimensionless distance between stringers

$\overset{\_}{b_{s}} = \frac{b_{s}}{l_{re}}$

versus the panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum feasible weight panel).

FIG. 51 is a plot showing dimensionless length of TPM shapes

${\overset{\_}{a} = \frac{a}{l_{re}}},{\overset{\_}{b} = \frac{b}{l_{re}}}$

versus the panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum feasible weight panel).

FIG. 52 is a plot showing dimensionless length of TPM shapes

${\overset{\_}{b} = \frac{b}{l_{re}}},{\overset{\_}{c} = \frac{c}{l_{re}}}$

versus the panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}}},$

material D16-T (minimum feasible weight panel).

FIG. 53 is a plot showing dimensionless thickness of U-shaped TPM

${\overset{\_}{\delta_{a}} = \frac{\delta_{a}}{l_{re}}},{\overset{\_}{\delta_{b}} = \frac{\delta_{b}}{l_{re}}},{\overset{\_}{\delta_{c}} = \frac{{\overset{\_}{\delta}}_{c}}{l_{re}}}$

versus panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}^{2}}},$

material D16-T (minimum feasible weight panel).

FIG. 54 is a plot showing dimensionless thickness of TPM

${\overset{\_}{\delta_{b}} = \frac{\delta_{b}}{l_{re}}},{\overset{\_}{\delta_{c}} = \frac{\delta_{c}}{l_{re}}}$

versus panel stress factor

${\overset{\_}{q_{p}} = \frac{q_{p}}{l_{re}^{2}}},$

material D16-T (minimum feasible weight panel).

2.5 Determining Optimum Inclination Angle of U-Shaped TPM Lateral Weband Optimum Pitch of Rivets for TPM Fastening to Skin

FIG. 55 is a stringer panel with U-shaped TPM possessing inclinedlateral web.

For the best utilization of the skin within the U-shaped TPM, thelateral webs of the latter are inclined to reach the equality ofb_(sk2)=b_(re).

The optimum inclination angle of the lateral web is:

$\begin{matrix}{{\alpha = {\arcsin \frac{{\overset{\_}{b}}_{sk} - \overset{\_}{a} - \overset{\_}{c}}{2}}},} & \left( {2.5{.1}} \right)\end{matrix}$

where b _(sk) is determined from equation (2.2.2).

For the minimum weight panel, σ_(s)=σ_(0.2); for the minimum feasibleweight panel, σ_(s)=σ_(n). Due to design restrictions, the angle αshould be 0≤α≤60°.

With this, the pitch of stringer installation is determined from theequation

b _(s) =b _(sk) +c+ā+2· b ·sin α  (2.5.2),

where b_(sk) is determined from (2.2.2).

In (2.5.2), the dimensions of the profile ā, b, c are dependent on theTPM stress factor P _(S), which, in its turn, is dependent on b _(s)(2.2.3). Therefore, the distance between stringers (pitch of stringerinstallation) is sought for through consecutive approximation approach.In the first approximation it is assumed that b _(s)=2·b_(re).

For the panels of minimum and least feasible weight, the optimum pitcht_(riv) shall be selected for TPM fastening to the skin.

The pitch t_(riv) is determined basing on equality of the overallstability critical stress for the skin between the rivets σ_(cr.riv) andthe stringer critical stress σ_(s), that is

$\begin{matrix}{{\sigma_{{cr} \cdot {riv}} = \sigma_{s}},{where}} & \left( {2.5{.3}} \right) \\{{\sigma_{{cr} \cdot {riv}} = {\frac{c \cdot \pi^{2} \cdot E}{\left( {{\overset{\_}{t}}_{riv}\text{/}{\overset{\_}{i}}_{sk}} \right)^{2}} \cdot \eta_{\sigma}}},} & \left( {2.5{.4}} \right)\end{matrix}$

ī_(sk)=δ/√{square root over (12)} is the skin radius of gyration.

Solving (2.5.3) accounting for (2.5.4) with respect to t _(riv), we get

$\begin{matrix}{{\overset{\_}{t}}_{riv} = {{{\overset{\_}{i}}_{sk} \cdot \pi \cdot \sqrt{\frac{c \cdot E \cdot \eta_{\sigma}}{\sigma_{s}}}} = {\frac{\delta}{2} \cdot \pi \cdot {\left( \frac{c \cdot E \cdot \eta_{\sigma}}{3 \cdot \sigma_{s}} \right)^{\frac{1}{2}}.}}}} & \left( {2.5{.5}} \right)\end{matrix}$

The equation (2.5.5) holds for both theoretical minimum and minimumfeasible weight panels. For the theoretical minimum weight panel,σ_(s)=σ_(0.2), while for the minimum feasible weight panel σ_(s)=σ_(n);for both panels, the ductility factor η_(σ) is calculated throughequation (1.2.2).

Should the pitch of rivets exceed the optimum one (2.5.5), the averagestress value in the panel skin (2.1.5) would decrease. Then thecross-section dimensions, hence the panel weight value, would increase.

The absolute dimensions of the TPM shape could prove not meet theconstructive restrictions; in this case, the dimensionless TPMparameters a₁=c/b; a₂=b/a; a₃=δ_(c)/δ_(b); a₄=δ_(b)/δ_(a) shall bealtered in compliance with the method stipulated in section 1. Withthis, the panel weight increases, though by the minimum feasible value.

2.6 Comparative Analysis of Weight Savings for Stringer Panels

Presented is investigation of possible weight savings for compressedstringer panels of a real torsion wing box, said panels designed by theiterative method. Benchmark data from these panels is utilized todetermine weight savings from method outlined in 2.3. With this purpose,compressed stringer panels designed by the iterative method are comparedwith calculated optimum weight compressed stringer panels using themethod outlined in 2.3.

The benchmark data were obtained for panels of torsion box wings of theairliner TU-154. For the purpose of comparative analysis, two panelsconsiderably differing in compressive loads are regarded. The panelswere designed for the compressive load q_(p) constituting, respectively,934 daN/cm and 555 daN/cm. The overall dimensions of panels are: lengthl_(s)=45 and 40 cm, respectively; widths B are 31.4 cm and 33.4 cm.

The longitudinal set (stringers) of these panels is manufactured out ofthe standard Z-shaped profiles: Pr 105-9 and Pr 105-1. The profiles werefastened to the skin with rivets. The profiles in the amount of five andsix pieces were evenly distributed across the panel width with the pitchb_(s)=70 mm and 58 mm, respectively.

Basing on the above benchmark data, optimum weight panels were designedin compliance with the technology outlined in 2.3.

FIG. 56 shows real and optimum design stringer panels.

For the purpose of visualization, portions of cross-sections of the realand optimum design panels are represented superimposed in FIG. 56.

Dimensions of profiles, skin thickness values, pitches of stringerinstallation, stresses and cross-section areas for real and optimumdesign panels are summarized in the Table 2.2 (FIG. 57).

Design of optimum weight panels using method in 2.3 decreased panelweights by 19.1% and 19.2%, respectively.

3.0 Design of Optimum Weight Rib-Reinforced Stringer Panel

For explanation purposes, analysis scheme for selection of optimumparameters of rib-reinforced stringer panel, (TPM installedlongitudinally and transversally) and calculations of these parametersare presented.

It has been noted in that “the exact solution of this kind of problemsis excessively intricate, and the effect of ribs is ratherinsignificant”. In the present section, we have managed to reduce thismulti-parameter problem to two variable parameters only, namely, thestress in the panel and the thickness of the skin, and to demonstratethe predominant role of ribs onto formation of the panel weight optimum.

The problems in sections 3 and 4 are solved in absolute values, whereasin sections 1 and 2 it is carried out in the dimensionless form. As thetransition from dimensionless values to absolute ones is ratherstraightforward, references to section 1 and 2 equations are given insections 3 and 4 without additional explanations.

FIG. 58 shows a rib-reinforced stringer panel.

FIG. 59 shows scheme for determining of the critical rigidity of ribs.

FIG. 60 shows scheme for determining of the rigidity of ribs.

FIG. 61 is a plot of dimensionless weight of the stringer panel

${{\overset{\_}{G}}_{p.s} = \frac{G_{p.s}}{l \cdot \gamma}},$

ribs

${\overset{\_}{G}}_{r} = \frac{G_{r}}{l \cdot \gamma}$

and rib-reinforced stringer panel G _(p.r)=G _(p.s)+G _(r) versus panelstress σ, material D16-T.

FIG. 62 is a plot of dimensionless weight of the rib-reinforced stringerpanel

${\overset{\_}{G}}_{p.s} = \frac{G_{p.s}}{l \cdot \gamma}$

versus distance between ribs l_(r), material D16-T.

FIG. 63 is a plot of dimensionless weight of the rib-reinforced stringerpanel

${\overset{\_}{G}}_{p.r} = \frac{G_{p.r}}{l \cdot \gamma}$

versus, skin thickness δ, material D16-T.

FIG. 64 is a plot of dimensionless weight

${\overset{\_}{G}}_{p.r} = \frac{G_{p.r}}{l \cdot \gamma}$

of the rib-reinforced stringer panel versus panel stress σ with varyingrib rigidity factor.

FIG. 65 is a plot of dimensionless weight

${\overset{\_}{G}}_{p.r} = \frac{G_{p.r}}{l \cdot \gamma}$

of the rib-reinforced stringer panel versus panel stress σ with varyingrib shape factor f_(r), material D16-T.

FIG. 66 is a plot of skin thickness δ of the rib-reinforced stringerpanel versus linear compressive force q_(p).

FIG. 67 is a plot of distance between stringers b_(s) of therib-reinforced stringer panel versus linear compressive force q_(p).

FIG. 68 is a plot of distance between ribs l_(r) of the rib-reinforcedstringer panel versus linear compressive force q_(p).

FIG. 69 is a plot of characteristic dimensions of stringer shape a, b ofthe rib-reinforced stringer panel versus linear compressive force q_(p).

FIG. 70 is a plot of characteristic thicknesses of stringer shape δ_(a),δ_(b) of the rib-reinforced stringer panel versus linear compressiveforce q_(p).

FIG. 71 is a plot of optimal stress σ of the rib-reinforced stringerpanel versus linear compressive force q_(p).

FIG. 72 is a plot of dimensionless weight

${\overset{\_}{G}}_{p.r} = \frac{G_{p.r}}{l \cdot \gamma}$

of the rib-reinforced stringer panel versus linear compressive forceq_(p).

Design Procedure for a Rib-Reinforced Stringer Panel

The discussed method is subdivided into the following stages:formulation of the problem, selection of the force pattern, determiningof the required moment of inertia for a rib and derivation of theanalytical expression for the rib-reinforced stringer panel.

The procedure is applicable for rib-reinforced stringer panel withframe- and beam-type ribs. We consider the panel with U-shaped stringershaving six variable dimensions, viz. a, δ_(a), b, δ_(b), c, δ_(c) andwith frame-type Z-shaped ribs having four variable parameters: b_(r),δ_(b.r), c_(r), δ_(c.r).

Depending on the shape of TPM (stringer or rib), the number of theirvariable dimensions can be different. Solution of this problem, as itwill be shown below, does not depend on the number of dimensions of thestringer or rib shape.

Formulation of the Problem

The problem in designing of the rib-reinforced stringer panel isformulated as follows: cross-section dimensions of the panel aredetermined, that is, stringer shape dimensions, rib shape dimensions,pitch of stringer installation b_(s), pitch of ribs installation l_(r),skin thickness δ basing on the overall panel minimum weight condition(FIG. 58). With this, the benchmark parameters are considered paneloverall dimensions B and l, linear compressive load q_(p) and the panelmaterial.

It is known from the stringer panel calculations that given the valuesof stress σ, skin thickness δ and panel length (equal to the pitch ofrib installation l_(s)=l_(r)), one can derive the optimal dimensions ofthe panel longitudinal set cross-section. Rib shape dimensions(transversal set) are determined by its bending rigidity E·J_(r), thatis, given the material E, by the rib shape moment of inertia J_(r). Themethod for determining of the shape dimension given its moment ofinertia is presented in section 3.2.

Therefore, designing of a rib-reinforced stringer panel is reduced todetermining of the panel stress σ, skin thickness δ, rib moment ofinertia J_(r) and the pitch of rib installation l_(r). It will be shownlater that the parameters J_(r) and l_(r) could be expressed throughparameters σ and δ. So, in the final analysis, designing of therib-reinforced stringer panel is reduced to finding the two variableparameters, panel stress σ and skin thickness δ.

Selection of the Scheme of Calculations

The rib-reinforced stringer panel is a complex spatial structure. Thefollowing calculation scheme is adopted for optimization of theparameters of the former: the stringers are considered to be isolatedrods on elastic supports. The elastic supports are considered to be theribs (FIG. 59). The ribs are regarded as double-support hinged beamsleaning on longerons. The skin participates in the panel operationacross the reduced width along with stringers and ribs. The worstoperating conditions will be experienced by the stringer located acrossthe panel width in the middle of the latter, as it would have the leastrigidity of elastic supports (for this stringer, the ribs have themaximum deflection). Assuming that destruction of a single stringerentails destruction of the panel as a whole, we assume the basicstringer to be the one located across the panel width in its middle.

Due to determining of the actual rigidity of the rib throughcalculations being too complicated, we use the adjustment factor, whichis to be obtained from the results of testing of the existing wingstructures. We will show below that the adjustment factor only slightlyaffects the optimal panel weight.

3.1 Determining of the Required Panel Moment of Inertia

Provided relatively high rigidity of elastically moving supports r_(r)(FIG. 60), loss of stability of the stringer occurs as if the supportwere absolutely rigid. The least support rigidity enabling this type ofstability loss is called the critical rigidity.

The critical rigidity of supports is determined through the followingexpression:

$\begin{matrix}{r_{r} = {\frac{m \cdot P_{s}}{\beta \cdot l} = {\frac{P_{s}}{\beta \cdot l_{r}}.}}} & \left( {3.1{.1}} \right)\end{matrix}$

Here, δ is the coefficient depending of the number of spans m. With thenumber of spans m exceeding 10, the coefficient δ would be the constantequal to 0.255; with number of spans being less, this coefficient grows.We assume δ=0.255, providing for the strength margin.

We further assume the ribs rigidity to be equal to the critical supportrigidity r_(r).

Therefore, the loss of the overall stringer stability could beconsidered at the length l_(r) (FIG. 60) according to Euler formula:

$\begin{matrix}{P_{s} = {\frac{c \cdot \pi^{2} \cdot E \cdot J_{s}}{l_{r}^{2}} \cdot {\eta_{\sigma}.}}} & \left( {3.1{.2}} \right)\end{matrix}$

Ductility factor η_(σ) in (3.1.2) for σ_(s)>σ_(pr) is derived throughformula (1.2.6) replacing therein σ_(cr.tot(η) _(σ) ₌₁₎ with σ_(s(η)_(σ) ₌₁₎.

Substituting (3.1.2) into (3.1.1), we obtain the critical rigidity ofthe rib as:

$\begin{matrix}{r_{r} = {\frac{c \cdot \pi^{2} \cdot E \cdot J_{s}}{\beta \cdot l_{r}^{3}} \cdot {\eta_{\sigma}.}}} & \left( {3.1{.3}} \right)\end{matrix}$

The sought-for required rib moment of inertia J_(r.rq) is determinedfrom the condition of equality of rib critical rigidity and rigidity ofthe rib considered as double-support beam.

The rigidity of the rib considered as double-support beam loaded withthe unit force P_(r) is expressed by the following equation (FIG. 60):

$\begin{matrix}{{r_{r} = {K_{ex} \cdot \frac{48 \cdot E \cdot J_{r.{rq}}}{B^{3}}}},} & \left( {3.1{.4}} \right)\end{matrix}$

K_(ex) is the experimental factor accounting for the actual rigidity ofthe rib within the wing; K_(ex)≥1.

Equating the critical rigidity (3.1.3) and rib rigidity (3.1.4), weobtain the required rib moment of inertia providing for the criticalrigidity of the support:

$\begin{matrix}{J_{r.{rq}} = {\frac{c \cdot \pi^{2} \cdot B^{2} \cdot J_{s}}{48 \cdot K_{ex} \cdot \beta \cdot l_{r}^{3}} \cdot {\eta_{\sigma}.}}} & \left( {3.1{.5}} \right)\end{matrix}$

In (3.1.5), the pitch of rib installation l_(r) is not determined. Letus determine it through variable parameters σ and δ.

Solving (1.3.7) with respect to l_(s)=l_(r), we get:

$\begin{matrix}{l_{r} = {\left\lbrack \frac{\pi^{4} \cdot E^{3} \cdot c^{2} \cdot P_{s}^{2} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{5}}{\sigma_{s}^{5}} \right\rbrack^{1/4}.}} & \left( {3.1{.6}} \right)\end{matrix}$

Here, the force per one stringer P_(s) is determined through (2.2.3);the pitch of stringer installation b_(s) in (2.2.3) is determinedthrough equations (2.2.2) or (2.5.2).

The stringer moment of inertia J_(s) in (3.1.5) is determined fromequations (1.3.3), (1.3.4) and (2.2.3).

Determining the Panel Weight

The panel weight is equal to the total of the weight of the longitudinal(stringers with skin) and transversal (ribs) sets.

$\begin{matrix}{G_{r \cdot {p.}} = {{\frac{q_{p} \cdot B}{\sigma} \cdot l \cdot \gamma} + {\frac{l}{l_{r}} \cdot F_{r} \cdot B \cdot \gamma \cdot {\left( {\frac{q_{p}}{\sigma} + \frac{F_{r}}{l_{r}}} \right).}}}} & \left( {3.1{.7}} \right)\end{matrix}$

Let us express the panel weight through the two variable parameters σand δ. For this, it is sufficient to express through these parametersthe ratio

$\frac{F_{r}}{l_{r}}.$

The rib area is determined according to the formula

$\begin{matrix}{{F_{r} = \left( \frac{J_{r}}{f_{r}} \right)^{\frac{1}{2}}},} & \left( {3.1{.8}} \right)\end{matrix}$

f_(r) is the rib shape factor.

The higher is the shape factor f_(r), the less is the weight of the rib.

Usually, the shape factor is determined after the rib strengthcalculations. Therefore, it would be better to determine it basing onthe analysis of the existing structures. It will be shown in thefollowing section how does the shape factor f_(r) influence the optimalpanel weight.

Substituting the expression for the rib moment of inertia (3.1.5) into(3.1.8), we obtain the area of the rib shape as a function of thestringer moment of inertia J_(s) and pitch of rib installation l_(r):

$\begin{matrix}{F_{r} = {\left\lbrack \frac{c \cdot \pi^{2} \cdot B^{3} \cdot J_{s} \cdot \eta_{\sigma}}{48 \cdot K_{ex} \cdot \beta \cdot f_{r} \cdot l_{r}^{3}} \right\rbrack^{\frac{1}{2}}.}} & \left( {3.1{.9}} \right)\end{matrix}$

The stringer shape moment of inertia (1.4.3) can be written, taking intoaccount (1.3.3) and (1.3.4), as:

$\begin{matrix}{{J_{s} = {\alpha \cdot \left( \frac{P_{s}^{5} \cdot l_{r}^{2}}{\sigma^{5} \cdot \eta_{\sigma}^{1/2}} \right)^{3}}},} & \left( {3.1{.10}} \right)\end{matrix}$

where the factor

${\alpha = \frac{A_{1}^{3} \cdot A_{2} \cdot A_{4}}{\pi^{2/3} \cdot c^{1/3}}},$

A₁, A₂, A₄ being the known values.

Substituting (3.1.10) into (3.1.9), we get the rib shape area as:

$\begin{matrix}{{F_{r} = {\phi \cdot \alpha^{1/2} \cdot \frac{P_{s}^{5/6} \cdot B^{3/2} \cdot \eta_{\sigma}^{1/3}}{\sigma^{5/6} \cdot l_{r}^{7/6}}}},} & \left( {3.1{.11}} \right)\end{matrix}$

where the factor

$\phi = {\sqrt{\frac{c \cdot \pi^{2}}{48 \cdot K_{ex} \cdot \beta \cdot f_{r}}}.}$

Solving (3.1.11) with respect to the ratio

$\frac{F_{r}}{l_{r}}$

and taking into account (3.1.6), we get:

$\begin{matrix}{{\frac{F_{r}}{l_{r}} = {\frac{\alpha^{1/2} \cdot \phi}{\psi} \cdot \frac{B^{3/2} \cdot \sigma^{15/8} \cdot \eta_{\sigma}^{1/16}}{P_{s}^{1/4} \cdot E^{13/8}}}},} & \left( {3.1{.12}} \right)\end{matrix}$

where the factor ψ=c^(13/12)·π^(13/6)·A^(65/24)

Substituting into (3.1.12) the expression for the force per one stringer(2.2.3), taking into account (2.2.2) and (2.3.7), we finally obtain:

$\begin{matrix}{{\frac{F_{r}}{l_{r}} = {\lambda \cdot \frac{B^{3/2} \cdot \sigma^{2}}{\left\lbrack {\delta \cdot \left( {q_{p} - {\delta \cdot \sigma}} \right) \cdot E^{7} \cdot \eta_{\sigma}^{1/4}} \right\rbrack^{1/4}}}},} & \left( {3.1{.13}} \right)\end{matrix}$

where λ is the constant defined by the expression

$\lambda = \left( \frac{A_{1}^{3} \cdot A_{2} \cdot A_{4}}{{48 \cdot 1},{9^{1/2} \cdot c^{3/2} \cdot A^{65/12} \cdot K_{ex} \cdot \pi^{3} \cdot \beta \cdot f_{r} \cdot K_{re}}} \right)^{\frac{1}{2}}$

Here: K_(re)=1 for TPM shape forming with the skin sheet an opencross-section configuration, K_(re)=2 for TPM shape forming with theskin sheet a close cross-section configuration.

Substituting (3.1.13) into (3.1.7), we obtain the analytical expressionfor the panel weight as a function of the two variable parameters σ andδ:

$\begin{matrix}{G_{r.p} = {\gamma \cdot l \cdot {\left\{ {\frac{q_{p} \cdot B}{\sigma} + {\lambda \cdot \frac{B^{5/2} \cdot \sigma^{2}}{\left\lbrack {\delta \cdot \left( {q_{p} - {\delta \cdot \sigma}} \right) \cdot \eta_{\sigma}^{1/4} \cdot E^{7}} \right\rbrack^{1/4}}}} \right\}.}}} & \left( {3.1{.14}} \right)\end{matrix}$

In the resulting expression for the weight (3.1.14), the first bracketedvalue determines the dimensionless weight of stringers with skin G_(p.s)(the weight of the longitudinal set), the second one determines theweight of ribs G_(r) (the weight of transversal set). It follows fromthe formula (3.1.14) that with increasing panel stress σ, the weight ofthe longitudinal set G_(p.s) diminishes, whereas the weight of thetransversal set G_(r) increases. It follows therefrom that the panelweight G_(p.r) has the optimum with respect to the stress σ. It alsofollows from consideration of the denominator of the second expressionin the formula (3.1.14) that the panel weight has an optimum withrespect to the skin thickness δ as well. The pitch of rib installation(3.1.6) is a function of the two variable parameters σ and δ, therefore,the panel weight has an optimum in respect of the pitch of ribinstallation, quod erat demonstrandum (that which was to bedemonstrated).

Combinations of the variable parameters σ and β corresponding to theminimum weight calculated through the formula (3.1.14) are the optimalparameters sought for. To determine these, the scanning method has beenemployed enabling finding the weight global minimum. The algorithm fordetermining the optimal parameters of the rib-reinforced stringer panelgiven the analytical expression for the weight value is given in section3.3.

Determined in the algorithm are the geometric characteristics ofstringer and rib shape staking into account their joint operation withthe skin and the optimal value of the side webs inclination angle forthe U-shaped stringer (2.5.1). The formulae for geometricalcharacteristics for stringer and rib with the reduced skin are presentedin the algorithm 3.3.

3.2 Procedure for Determining the TPM Shape Dimensions Given its KnownGeometrical Characteristics

Let us demonstrate how the rib shape dimensions can be found given itsmoment of inertia J_(r,rq.) (3.1.5).

Assume that in determination of the rib shape factor f_(r) (3.1.8), theshape is selected with the dimensions b_(r)′, δ_(b.r)′, c_(r)′, δ_(c.r)′and with the local stability critical stress σ_(cr.l)′.

Then the factor of rib shape dimensions change r shall be determinedfrom the expression

$\begin{matrix}{{\psi_{r} = \left( \frac{J_{r.{rq}}}{J_{r}^{\prime}} \right)^{1/4}},} & \left( {3.2{.1}} \right)\end{matrix}$

where: J_(r.rq) is the required optimal value of the rib moment ofinertia;

J_(r)′ is the moment of inertia for the selected rib shape withdimensions b_(r)′, δ_(b.r)′, c_(r)′, δ_(c.r)′

The sought-for optimal dimensions of the rib shape are determined fromthe following expressions:

b _(r)=ψ_(r) ·b′ _(r) , c _(r)=ψ_(r) ·c′ _(r),

δ_(b)=ψ_(r)·δ′_(b), δ_(c.r)=ψ_(r)·δ′_(c.r)  (3.2.2).

The rib shape dimensions found from the condition of its criticalrigidity shall be increased after the rib strength calculations. Todetermine rib shape dimensions from the strength condition, let us usethe above approach. Provided employing this approach, the localstability critical stress σ_(cr.l)′ and the shape factor f_(r) adoptedin the stability calculations do not change.

The moment of resistance of the rib shape is determined from thestrength condition:

$\begin{matrix}{W = {\frac{M_{r}}{\sigma_{{cr}.l}^{\prime}}.}} & \left( {3.2{.3}} \right)\end{matrix}$

Here, M_(r) is the maximum bending torque in the rib cross-section;

σ_(cr.l)′ is the critical local stability stress for the rib shape withdimensions b_(r)′, δ_(b.r)′, c_(r)′, δ_(c.r)′.

Then the factor of rib shape change found from the strength conditionshall be determined from the formula:

$\begin{matrix}{{\psi_{r} = \left( \frac{W_{r}}{W_{r}^{\prime}} \right)^{1/3}},} & \left( {3.2{.4}} \right)\end{matrix}$

W_(r)′ is the moment of resistance of the selected rib shape withdimensions b_(r)′, δ_(b.r)′, c_(r)′, δ_(c.r)′.

The rib shape dimensions are determined from expressions (3.2.2).

If apart from the bending torque M_(r) the rib is loaded also by anaxial force P_(ax)(for example, a frame rib), then the shape changefactor can be determined from the expression:

$\begin{matrix}{{{{\pm \frac{M_{r}}{\psi^{3} \cdot W_{r}^{\prime}}} + \frac{P_{ax}}{\psi^{2} \cdot F_{r}^{\prime}}} \leq \sigma_{{cr}.l}^{\prime}},} & \left( {3.2{.5}} \right)\end{matrix}$

W_(r)′ and F_(r)′ being, respectively, moment of resistance and area ofthe rib shape with dimensions b_(r)′, δ_(b.r)′, c_(r)′, δ_(c.r)′.

3.3 Calculations of Optimal Parameters for Rib-Reinforced Stringer Panel

Presented in this section is the algorithm to obtain the optimalparameters. Some computation results are presented in the form ofdiagrams required for analysis of the results obtained, as well as ofplots of optimal parameters for the rib-reinforced stringer panel versusthe linear compressive force.

Algorithm

(1) The benchmark data are set:

the linear force compressing the panel q_(p);

mechanical characteristics of the panel material;

panel width B;

panel length l;

stringer shape factors a₁, a₂, a₃, a₄, A₁, A₂, A₃, A₄;

benchmark dimensions, geometrical characteristics and local stresscritical stability of the rib b′, δ_(b.r)′, c′, δ_(c.r)′, F_(r)′,W_(r)′, J_(r)′, f_(r), σ_(cr.l)′;

the coefficient in the rib rigidity expression K_(ex);

the factor accounting for the condition of the stringer leaning on therib c;

the factor depending on the number of spans β;

the factor depending on openness or closeness of the configurationformed by the TPM with the sheet, K_(re).

(2) The cycles are specified in the following variable parameters:

(a) skin thickness δ; the series of values in compliance with theexisting standards is adopted, for example, δ=1; 1.2; 1.5; 2; 2.5; 3;3.5; 4; 4.5; 5 [mm];

(b) panel stress σ; the range of variation is adopted

σ=(500−σ_(0.2))[daN/cm²].

(3) The ductility factor is calculated

η_(σ)=1 provided σ≤σ_(pr);

$\eta_{\sigma} = {\frac{\sigma}{\sigma_{pr}} \cdot \left( \frac{\sigma_{0} - \sigma}{\sigma_{0} - \sigma_{pr}} \right)^{2}}$

provided σ>σ_(pr).

The algorithm for panel parameters optimization follows for the knownanalytical value for the panel weight (3.1.14).

The following coefficient is calculated:

$\lambda = {\frac{A_{1}^{3} \cdot A_{2} \cdot A}{48 \cdot 1.9^{1/2} \cdot c^{3/2} \cdot A^{65/12} \cdot K_{\ni} \cdot \pi^{3} \cdot \beta \cdot f_{r} \cdot K_{re}}.}$

The minimum panel weight G_(p.r) is determined through selection ofoptimal combinations of the variable parameters σ and δ:

$G_{r,p} = {\left\{ {\frac{q_{p} \cdot B}{\sigma} + {\lambda \cdot \frac{B^{5/2} \cdot \sigma^{2}}{\left\lbrack {\delta \cdot \left( {q_{p} - {\delta \cdot \sigma}} \right) \cdot E^{7} \cdot \eta_{\sigma}^{1/4}} \right\rbrack^{1/4}}}} \right\} \cdot l \cdot {\gamma.}}$

The optimal reduced skin width is determined:

$b_{re} = {1.9 \cdot \delta \cdot {\left( \frac{E}{\sigma} \right)^{1/2}.}}$

The optimal pitch of stringer installation is determined:

for shapes forming with the sheet an open configuration b_(s)=2·b_(re),

for those forming with the sheet a close configuration b_(s)=b_(re).

The force per a stringer is calculated:

P _(s) =b _(s)·(q _(p)−δ·σ).

The optimal pitch of rib installation is determined:

$l_{r} = {\left\lbrack \frac{\pi^{4} \cdot E \cdot c^{2} \cdot P_{s}^{2} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{5}}{\sigma^{5}} \right\rbrack^{1/4}.}$

The optimal stringer shape dimensions are determined:

${a = {\left( \frac{P_{s} \cdot l_{r}}{\pi \cdot \sigma \cdot c^{1/2}} \right)^{1/3} \cdot \frac{A_{1}}{\eta_{\sigma}^{1/2}}}};$${\delta_{a} = {\left( \frac{\pi \cdot P_{s}^{2} \cdot c^{1/2}}{\sigma \cdot l_{r}} \right)^{1/3} \cdot A_{2} \cdot \eta_{\sigma}^{1/2}}};$b = a ⋅ a₂, c = b ⋅ a₁, δ_(b) = δ_(a) ⋅ a₄, δ_(c) = δ_(b) ⋅ a₃.

The stringer moment of inertia is calculated:

J _(s) =a ³·δ_(a) ·A ₄.

The required rib shape moment of inertia is determined:

$J_{r,p} = {\frac{c \cdot \pi^{2} \cdot J_{s} \cdot B^{3} \cdot \eta_{\sigma}}{48 \cdot \beta \cdot l_{r}^{3}}.}$

The factor of rib shape change is calculated:

$\psi_{r} = {\left( \frac{J_{r.p.}}{J_{r}^{\prime}} \right)^{1/4}.}$

Optimal dimensions of the rib shape are determined:

b _(r)=ψ_(r) ·b′ _(r); δ_(b.r.)=ψ_(r)·δ′_(b.r) ; c _(r)=ψ_(r) ·c′ _(r);δ_(c.r)=ψ_(r)·δ′_(c.r).

Presented below is the refined algorithm for the rib-reinforced stringerpanel optimization accounting for the influence of the skin onto thebending rigidity of the longitudinal and transversal sets, as well asfor the inclination angle of the U-shaped profile lateral webs α.

Steps (1), (2), (3) remain the same as in the above algorithm.

The reduced skin width is determined:

$b_{re} = {1.9 \cdot \delta \cdot {\left( \frac{E}{\sigma} \right)^{1/2}.}}$

The pitch of stringer installation is determined:

for shapes forming with the sheet an open configuration b_(s)=2·b_(re),for those forming with the sheet a close configuration b_(s)=b_(re).

The force per one stringer is determined:

P _(s) =b _(s)·(q _(p)−δ·σ).

The pitch of rib installation is determined:

$l_{r} = {\left\lbrack \frac{\pi^{4} \cdot E \cdot c^{2} \cdot P_{s}^{2} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{5}}{\sigma^{5}} \right\rbrack^{1/4}.}$

Stringer shape dimensions are determined:

${a = {\left( \frac{P_{s} \cdot l_{r}}{\pi \cdot \sigma \cdot c^{1/2}} \right)^{1/3} \cdot \frac{A_{1}}{\eta_{\sigma}^{1/2}}}};$${\delta_{a} = {\left( \frac{\pi \cdot P_{s}^{2} \cdot c^{1/2}}{\sigma \cdot l_{r}} \right)^{1/3} \cdot A_{2} \cdot \eta_{\sigma}^{1/2}}};$b = a ⋅ a₂, c = b ⋅ a₁, δ_(b) = δ_(a) ⋅ a₄, δ_(c) = δ_(b) ⋅ a₃.

For the U-shaped profile, the web inclination angle is determined:

$\alpha = {{arc}\; \sin {\frac{b_{re} - a - c}{2 \cdot b}.}}$

If α≥60°, then it is assumed that α=60°; if α<0°, then it is assumedthat α=0°. Then the value of b, is calculated:

b _(s) =c+a+b _(re)+2·b·sin α.

If 0<α≤60°, then steps 6 to 8 are reiterated; then branching to step 12is carried out.

The stringer shape moment of inertia is determined:

J _(s) =a ³·δ_(a) ·A ₄·cos² α.

The coordinate of the stringer shape neutral axis is determined:

For U-shaped profile:

${Y_{s} = \frac{a \cdot \left( {1 + {a_{2} \cdot a_{4}}} \right)}{1 + {2 \cdot \left( {{a_{2} \cdot a_{4}} + {a_{1} \cdot a_{2} \cdot a_{3} \cdot a_{4}}} \right)}}},$

For Z-shaped profile:

${Y_{s} = \frac{a}{2}},$

For L-shaped profile:

$Y_{s} = {\frac{a}{2 \cdot \left( {1 + {a_{1} \cdot a_{2}}} \right)}.}$

The stringer shape area is determined:

for shapes forming with the sheet a closed configuration (U-shapedprofile):

F _(s) =a·δ _(a)·(1+2·a ₂ ·a ₄+2·a ₁ ·a ₂ ·a ₃ ·a ₄),

for shapes forming with the sheet an open configuration:

(Z-shaped profile) F_(s)=a·δ_(a)·(1+2·a₁·a₂),

(L-shaped profile) F_(s)=a·δ_(a)·(1+a₁·a₂).

The coordinate of the stringer shape neutral axis with scaled (reduced)skin is determined:

$Y_{s.{re}} = {\frac{F_{s} \cdot Y_{s}}{F_{s} + {b_{s} \cdot \delta}}.}$

The scaled (reduced) moment of inertia of the stringer shape isdetermined:

J _(s.re) =J _(s) +F _(s)·(Y _(s) −Y _(s.re))² +b _(s) +δ·Y _(s.re) ².

The required rib shape moment of inertia is determined:

$J_{r.{rq}} = {\frac{c \cdot \pi^{2} \cdot J_{s.{re}} \cdot B^{3} \cdot \eta_{\sigma}}{48 \cdot K_{ex} \cdot \beta \cdot l_{r}^{3}}.}$

The coordinate of the rib shape neutral axis with reduced skin isdetermined (for Z-shaped profile):

$Y_{r.{re}} = {\frac{F_{r}^{\prime} \cdot b_{r}^{\prime}}{2 \cdot \left( {F_{r}^{\prime} + {b_{re} \cdot \delta}} \right)}.}$

Rib shape moment of inertia is determined:

J _(r) =J _(r.rq) −b _(re) ·δ·Y _(r.re).

The rib shape area is determined:

$F_{r} = {\left( \frac{J_{r}}{f_{r}} \right)^{1/2}.}$

The minimum panel weight G_(p.r) is determined through selection ofoptimal combinations of the variable parameters σ and δ:

$G_{r.p.} = {\gamma \cdot l \cdot B \cdot {\left( {\frac{q_{p}}{\sigma} + \frac{F_{r}}{l_{r}}} \right).}}$

The factor of rib shape change is calculated

$\psi_{r} = {\left( \frac{J_{r}}{J_{r}^{\prime}} \right)^{\frac{1}{4}}.}$

The optimal rib shape dimensions are determined:

b _(r)=ψ_(r) ·b′ _(r); δ_(b.r)=ψ_(r)·δ′_(b.r) ; c _(r)=ψ_(r) ·c′ _(r);δ_(c.r)=ψ_(r)·δ′_(c.r).

The panel parameters corresponding to the minimum weight are output.

The calculation results are presented as plots of two types: weightversus optimal parameters (FIGS. 61-65); optimal parameters versuslinear compressive force (FIGS. 66-72). The plots are presented forthree stringer shapes, materials D16-T and B95.

From the plot in FIG. 61 one can see the influence of the rib weightG_(r) onto the panel optimal weight G_(p.r). With the increase of thestress σ, the weight of the longitudinal set G_(p.s) falls smoothly,while the weight of ribs increases, first slowly and then sharply. Theoptimal skin thickness can be seen to constitute δ=3 mm. The total ribsweight G_(r) being rather small compared to that of the longitudinal setG_(p.s.), it is this value that determines the optimal panel weightG_(p.r.). Deviations from the optimal value of the panel weight entailthe sharp increase of the latter's weight.

From the plot in FIG. 62 one can see dependence of the optimal panelweight from the skin thickness. For the selected step of itsincrementing δ=1 mm, the optimal skin thickness is δ=3 mm. With the skinthickness more than δ=3 mm or less than this value, the panel weightincreases. It can also be seen from the plot in FIG. 62 that deviationfrom the optimal value of the skin thickness entails insignificantincrease of the panel weight. Therefore, varying the skin thickness δ,one can arrange iteration cycles in δ from an array of sheet thicknessvalues, selecting those according to the existing standards, similarlyto the way employed in the suggested algorithm.

Shown in the plot in FIG. 63 is the influence of the pitch of ribinstallation onto the panel weight. It follows from this plot thatdeviation from the optimal pitch of rib installation increases the panelweight relatively insignificantly; a twofold deviation from the optimalpitch of rib installation (for skin thickness δ=1, 2, 3 mm) increasesthe panel weight in the average by 10%.

Shown in the plot in FIG. 63 is the influence of the rib rigidity ontothe minimum panel weight. In the case of twofold increase of theadjustment factor of the rib rigidity K_(ex) (3.1.14), the minimum panelweight is reduced by 4.7%; increasing of K_(ex) by the factor of fourentails reduction of the minimum panel weight by 6.2%. An error in theaccuracy in determining the rib rigidity would entail a relatively smallincrease in the panel weight. This by no means cancels the requirementof obtaining as precise value of the rib rigidity factor K_(ex) aspossible.

The influence of the rib shape factor onto the minimum panel weight isshown in FIG. 65. It can be seen from the plot that with the rib shapefactor two fold increase, the minimum panel weight is reduced by 4.7%.It is advisable, other conditions being equal, to have the maximumpossible value of the rib shape factor; that is, the rib shape should beselected with its area as spaced as possible.

One can see from the plot in FIG. 72 that the dependence of the panelweight versus the linear force could be approximated fairly by astraight line. The panel manufactured out of a material with highermechanical properties possesses a smaller weight.

Given the value of the weight G _(p.r), at a certain value of linearcompressive force q_(p), it is possible to get an analytical expressionfor G_(p.r) as a function of q_(p) through the formula

${G_{p.r} = {{l \cdot \gamma \cdot q \cdot {tg}}\; \alpha}},{{{where}\mspace{14mu} {tg}\; \alpha} = {\frac{{\overset{\_}{G}}_{p.r}}{q_{p}}.}}$

4.0 Designing of Wing Torsion Box Compressed Members

Discussed in this section is method of design and calculation of optimalin weight cross-section dimensions of wing torsion box load-carryingmembers.

The method is divided into several stages. At the first stage (section4.1), the problem is formulated and the calculation scheme is selected.At the second stage (section 4.2), the basic equation for wing torsionbox weight is obtained at several simplifying assumptions; it is shownhere that the variable parameters of the basic equation of the torsionbox weight possess optimums. At the third stage (section 4.3), theproblem are solved with account for normal and tangential stressesacting on the torsion box members; the formulae for numerical solutionof the problem are presented. At the fourth stage (section 4.4) thealgorithm for wing optimal parameters selection is presented. Theproblem is solved without assumptions adopted in section 4.2: jointaction of normal and tangential stresses onto wing torsion box membersis accounted for; also accounted for is the effect of the skin onto thebending rigidity of longitudinal and transversal sets and optimal angleof inclination of U-shaped stringer lateral webs is determined. Itshould be emphasized that each of the stages employs the results of theprevious ones.

FIG. 73 plots aerodynamic profile of the wing with the specifiedposition of the torsion box.

FIG. 74 shows a wing design scheme.

FIG. 75 is a table showing coefficients K_(σ.w), K_(τ.w), in theformulae for web cell critical stresses.

FIG. 76 is a table showing factor θ versus web cell overall dimensions.

FIG. 77 is a table showing feasible coefficients for a stringer shape.

FIG. 78 is a table showing dimensions and geometrical characteristics ofrib shape.

FIG. 79 is a table showing dimensions and geometrical characteristics ofpillar shape.

4.1 Problem Formulation and Selection of the Calculation Scheme

The problem of design of load-carrying members of the torsion box isformulated as follows: the cross-section dimensions of load-carryingmembers of the torsion box are sought for basing on the criterion of theminimum weight.

The parameters to be determined:

-   -   stringer shape dimensions a, δ_(a), b, δ_(b), C, δ_(c) (U-shaped        stringer);    -   rib shape dimensions b_(r), δ_(b.r), c_(r), δ_(c.r) (Z-shaped        rib);    -   distance (pitch) between stringers b_(s), distance (pitch)        between ribs l_(r), skin thickness β (FIG. 74);    -   maximum thickness of the aerodynamic profile c_(max) (effective        torsion box height h_(ef)), thickness of vertical webs t (FIGS.        73, 74);    -   shape dimensions for pillars reinforcing the web b_(pi),        δ_(b.pi), c_(pi), δ_(c.pi) (L-shaped profile), distance between        pillars r.

Prescribed values are:

loads in the wing design sections: bending torque M, transversal forceQ, torsion torque M_(ts); distance between torsion box design sectionsl;wing aerodynamic profile with the given torsion box dimensions h_(w1),h_(w2), B (FIG. 73);

Given the wing aerodynamic profile, the optimal position of the torsionbox (dimensions h_(w1), h_(w2), B) is found by known method basing onthe criterion of obtaining the maximum building height of the torsionbox.

To determine the weight and optimal dimensions of the membercross-sections, the calculation scheme shown in FIG. 74 is employed. Asfollows from this calculation scheme, the top and bottom panels possessequal cross-section areas.

The maximum thickness of the wing aerodynamic profile c_(max) is relatedto the effective height h_(ef) through the coefficient

$\begin{matrix}{{c_{\max} = \frac{h_{ef}}{K_{ef}}},{{{where}\mspace{14mu} K_{ef}} = {0.85 - {0.7.}}}} & \left( {4.1{.1}} \right)\end{matrix}$

Lower values of K_(ef) correspond to “thicker” torsion box panels; thevalue K_(ef)=0.85 corresponds to zero thickness; it is assumed thatK_(ef)=0.75.

In the course of the iterative calculations the adopted value of K_(ef)can be always refined according to the equation

$\begin{matrix}{{K_{ef} = {K_{{ef}.{tot}.} - {0.985\frac{\delta_{p}}{c_{\max}}}}},} & \left( {4.1{.2}} \right)\end{matrix}$

δ_(p) being the effective plate thickness constant along the panelwidth;

$\begin{matrix}{{\delta_{p} = {\delta + \frac{F_{s}}{b_{s}}}},{K_{{ef}.{tot}.} = {\frac{1}{N \cdot c \cdot Z_{\max}} \cdot {\sum\left( {Z_{{top}.p}^{2} + Z_{{bot}.p}} \right)}}}} & {\left( {4.1{.3}} \right),}\end{matrix}$

Z being the ordinate of the profile measured from the torsion boxneutral axis, N the number of elementary area sites.

Heights of the torsion box webs are related to the maximum thickness ofthe wing aerodynamic profile c_(max) with the equation

h _(w) =K _(w) ·c _(max)  (4.1.4),

K_(w) being the factor, which is defined in the first approximationthrough the given torsion box web heights:

${K_{w\; 1} = \frac{h_{w\; 1}}{c_{\max}}},{K_{w\; 2} = {\frac{h_{w\; 2}}{c_{\max}}.}}$

The value of K_(w) can be always refined according to the found optimalvalue of c_(max).

The expression (4.1.4) taking into account (4.1.1) takes on the form:

h _(w) =K _(ef.w.) ·h _(ef)  (4.1.5),

K_(ef.w) being the given value,

$K_{{ef}.w.} = {\frac{K_{w}}{K_{ef}}.}$

The torsion torque is reacted by the torsion box as a whole. The designvalue of the torsion box cross-section area is

Ω_(d) =K _(ts)·Ω  (4.1.6),

K_(ts) being the factor determined by the ratio of the design area ofthe torsion box cross-section Ω_(d) to the given one Ω; in the firstapproximation, K_(ts)=1.The value of K_(ts) can be always refined according to the found optimalvalue of c_(max).

Vertical webs of the torsion box are assumed to be of the samethickness, or the ratio of thickness values for these should bespecified. Otherwise, each additional web thickness adds anothervariable parameter.

The torsion box members are selected basing on the stability condition;then the ribs are checked for strength.

For the selected wing calculation scheme, the bending torque is reactedby the wing panels at height h_(ef) and by webs at the height h_(w); thetransversal force is distributed proportionally to the bending rigidityof the webs and is reacted by the latter ones at the height h_(w); thetorsion torque is reacted by the torsion box cross-section as a whole.

4.2 Derivation of the Basic Equation for Wing Torsion Box Weight

In deriving the analytical expression for the basic equation for wingtorsion box weight it is assumed that the tangential force brought aboutby the transversal force and by the torsion torque affect the paneldimensions only slightly, and the normal force brought about by thebending torque affect the dimensions of the web and its reinforcingpillars slightly as well; therefore, the above effects shall beneglected.

The wing torsion box weight is equal to the total of the weight ofrib-reinforced stringer panels and pillar-reinforced vertical webs.

G=2·(G _(p.r) +G _(w))  (4.2.1),

where it is assumed that h_(w1)=h_(w2)=h_(w).

The expression for the rib-reinforced stringer panel weight compressedby the linear force q_(p) has the form (3.1.14).

The first member in (3.1.14) determines the weight of stringers, whilethe second the weight of ribs. It follows from (3.1.14) that the weightof the compressed panel is the function of two variable parameters,normal stress in panel a and skin thickness β. Let us emphasize that allpanel cross-section dimensions are expressed through parameters α and δ.

The linear compressive force q_(p) is expressed through the givenbending torque and the sought-for effective height of the torsion box asfollows:

$\begin{matrix}{q_{p} = {\frac{M}{B \cdot h_{ef}}.}} & \left( {4.2{.2}} \right)\end{matrix}$

Taking into account (4.2.2), the weight of the rib-reinforced stringerpanel (3.1.14) shall take on the form

$\begin{matrix}{G_{r \cdot p} = {\left\{ {\frac{M}{h_{ef} \cdot B} + {\lambda \cdot \frac{B^{\frac{5}{2}} \cdot \sigma^{2}}{\left\lbrack {\delta \cdot \left( {\frac{M}{h_{ef} \cdot B} - {\delta \cdot \sigma}} \right) \cdot E^{7} \cdot \eta_{\sigma}^{\frac{1}{4}}} \right\rbrack^{\frac{1}{4}}}}} \right\} \cdot l \cdot {\gamma.}}} & \left( {4.2{.3}} \right)\end{matrix}$

The weight of the rib-reinforced stringer panel (4.2.3) depends on threevariable parameters σ, β and h_(e)f

The weight of pillar-reinforced web appearing in (4.2.1) can be writtenas follows:

$\begin{matrix}{G_{w} = {\left( {t + \frac{F_{pi}}{r}} \right) \cdot K_{{ef}.w.} \cdot h_{ef} \cdot l \cdot {\gamma.}}} & \left( {4.2{.4}} \right)\end{matrix}$

It follows from (4.2.4) that the weight of pillar-reinforced web is thefunction of four variables: t, r, F_(pi), h_(ef) In its turn, the pillararea F_(pi) depends on pillar shape dimensions. For the L-shapedprofile, F_(pi) is the function of four parameters b_(pi), δ_(b.pi),c_(pi) and δ_(c.pi).

Determining of the optimal dimension for the cross-section of thepillar-reinforced web loaded by the flow of tangential forces is adistinct problem. In known studies of P.Kun and G.Hertel, this problemhad been considered for a web buckling due to shear. We shall not admitthe web buckling, as “it would be inexpedient to admit buckling due tooperation of stripes (longerons) for longitudinal-transversal bending”.

The optimal web cross-section dimensions shall be determined basing oncondition of its minimum weight. This approach enables to express theparameters t and r explicitly as a function of the given load, of themechanical properties of material and of the pillar shape area F_(pi).The pillar shape area shall be selected in the first approximationbasing on the analysis of the existing structures or basing onconstructive considerations. Therefore, the wing weight shall be afunction of three variable parameters, σ, δ and h_(ef).

The procedure to determine the optimal parameters of the web basing oncondition of the torsion box minimum weight along with selection of theoptimal parameters of the pillar is stipulated in section 4.4. The wingtorsion box weight shall be a function of four variable parameters σ, δ,h_(ef), and t.

The web strength condition has the form

τ_(cr.w)=τ_(w)  (4.2.5)

The critical tangential stress of web cell σ_(cr.w) is determined fromthe equation

$\begin{matrix}{\tau_{{cr}.w} = {\frac{K_{\tau \cdot w \cdot} \cdot E \cdot t^{2}}{r^{2}} \cdot {\sqrt{\eta_{\tau.w.}}.}}} & \left( {4.2{.6}} \right)\end{matrix}$

The factor K_(σ.w) is assumed the same as for the pivoted infinitelylong plate; it is assumed that r<h_(w); K_(τ.w)=4.8.

The ductility factor for shear η_(τ) can be obtained similarly to theductility factor for compression η_(σ).

Replacing a with τ in (1.2.5), we get

$\begin{matrix}{\eta_{\tau} = {\left\{ \frac{\left( \left\lbrack {\sqrt{{\tau_{pr} \cdot \left( {\tau_{0} - \tau_{pr}} \right)^{2}} + {4 \cdot \tau_{{cr}{({\eta_{\tau} = 1})}}^{2} \cdot \tau_{0}}} - {\tau_{pr}^{1/2} \cdot \left( {\tau_{0} - \tau_{pr}} \right)}} \right\rbrack^{2} \right)}{4 \cdot \tau_{{cr}{({\eta_{\tau} = 1})}}^{3}} \right\}.}} & \left( {4.2{.7}} \right)\end{matrix}$

The web ductility factor η_(τ) in (4.2.6) is determined as follows. Thevalue η_(τ)=1 is set, and the critical tangential stress τ_(ck.w(η) _(τ)₌₁₎ is calculated through (4.2.6). Should the value τ_(ck.w(η) _(τ)₌₁₎≤τ_(pr), the ductility factor η_(τ.w)=1; otherwise (τ_(ck.w(η) _(τ)₌₁₎>τ_(pr)), the calculated value τ_(kp.w(η) _(τ) ₌₁₎ is substitutedinto (4.2.7), whereby η_(τ.w) is found. Finally, the critical stressτ_(cr.w) is determined from the equation

τ_(ck.w)=τ_(ck.w(η) _(τ) ₌₁₎·√{square root over (η_(τ.w))}  (4.2.8).

The tangential stress in the web, taking into account (4.1.5) and(4.1.6) takes on the form

$\begin{matrix}{{\tau_{w} = {\frac{M_{ts}}{2 \cdot K_{ts} \cdot \Omega \cdot t} + \frac{Q_{{tr}.d}}{K_{{ef}.w} \cdot h_{ef} \cdot t}}},} & \left( {4.2{.9}} \right)\end{matrix}$

Q_(tr.d) being the transversal force in the design section of the webtaking into account its obliquity.

$\begin{matrix}{{Q_{{tr}.d} = {\frac{1}{2}\left( {Q - {\frac{M}{K_{{ef}.w} \cdot h_{ef}} \cdot \alpha_{w}}} \right)}},} & \left( {4.2{.10}} \right)\end{matrix}$

α_(w) being the obliquity angle of webs, the factor ½ before theparentheses taking into account consideration of two webs of the sameheight.

For various values of thickness and height of torsion box web, or incase of number of webs exceeding 2, the tangential forces acting ontoeach of the webs can be obtained from the known equations as functionsof parameters h_(w) and t.

Solving equation (4.2.5) taking into account (4.2.6) and (4.2.9) for theweb thickness t, we get

$\begin{matrix}{t = {\left\lbrack {\frac{r^{2}}{K_{\tau.w} \cdot E \cdot \sqrt{\eta_{\tau.w}}} \cdot \left( {\frac{M_{ts}}{2 \cdot K_{ts} \cdot \Omega} + \frac{Q_{{tr}.d}}{K_{{ef}.w} \cdot h_{ef}}} \right)} \right\rbrack^{1/3}.}} & \left( {4.2{.11}} \right)\end{matrix}$

Taking into account (4.2.1), the equation for the web weight takes onthe form

$\begin{matrix}{G_{w} = {K_{{ef}.w.} \cdot h_{ef} \cdot l \cdot \gamma \cdot {\left\{ {\left\lbrack \frac{r^{2}\left( {\frac{M_{ts}}{2 \cdot K_{ts} \cdot \Omega} + \frac{Q_{{tr}.d}}{K_{{ef}.w} \cdot h_{ef}}} \right)}{K_{\tau.w} \cdot E \cdot \sqrt{\eta_{\tau.w}}} \right\rbrack^{1/3} + \frac{F_{pi}}{r}} \right\}.}}} & \left( {4.2{.12}} \right)\end{matrix}$

The distance between pillars is determined from the equation

$\begin{matrix}{\frac{\partial G_{w}}{\partial r} = 0.} & \left( {4.2{.13}} \right)\end{matrix}$

Solving (4.2.13) for r taking into account (4.2.12), we get

$\begin{matrix}{r = {\left\lbrack \frac{\left( {\frac{3}{2}F_{pi}} \right)^{3} \cdot K_{\tau.w} \cdot E \cdot \sqrt{\eta_{\tau.w}}}{\frac{M_{ts}}{2 \cdot K_{ts} \cdot \Omega} + \frac{Q_{{tr}.d}}{K_{{ef}.w} \cdot h_{ef}}} \right\rbrack^{1/5}.}} & \left( {4.2{.14}} \right)\end{matrix}$

Substituting (4.2.14) into (4.2.11), we get the web thickness texpressed through the selected pillar shape area F_(pi) as

$\begin{matrix}{t = {\left\lbrack \frac{\frac{9}{4}{F_{pi}^{2} \cdot \left( {\frac{M_{ts}}{2 \cdot K_{ts} \cdot \Omega} + \frac{Q_{{tr}.d}}{K_{{ef}.w} \cdot h_{ef}}} \right)}}{K_{\tau.w} \cdot E \cdot \sqrt{\eta_{\tau.w}}} \right\rbrack^{1/5}.}} & \left( {4.2{.15}} \right)\end{matrix}$

The ductility factor η_(τ.w) appearing in the equations for determiningthe distance between pillars (4.2.14) and of web thickness (4.2.15) isdetermined as follows. η_(τ.w)=1 is assumed, after which calculated arethe distance between pillars r_((η) _(τ) ₌₁₎ through (4.2.14), the webthickness t_((η) _(τ) ₌₁₎ through (4.2.15) and the critical tangentialstress in the web τ_(ck.w(η) _(τ) ₌₁₎ through (4.2.6). Should it turnout that τ_(ck.w(η) _(τ) ₌₁₎≤σ_(pr), then η_(τ.w)=1, otherwise η_(τ.w)is calculated according to equation (4.2.7). The final values for r, tand τ_(cr.w) are determined accounting for the obtained value η_(τ.w).

Substituting into (4.2.4) the expressions for the web thickness (4.2.15)and the distance between pillars (4.2.14), we get the web weight as

$\begin{matrix}{G_{w} = {{\gamma \cdot l \cdot h_{ef}}{K_{{ef}.w} \cdot {{\left( {K_{\tau.w} \cdot E \cdot \sqrt{\eta_{\tau.w}}} \right)^{{- 1}/5}\left\lbrack {\frac{2}{3}{F_{pi}^{2}\left( {\frac{M_{ts}}{2{K_{ts} \cdot \Omega}} + \frac{Q_{{tr}.d}}{K_{{ef}.w} \cdot h_{ef}}} \right)}} \right\rbrack}^{1/5}.}}}} & \left( {4.2{.16}} \right)\end{matrix}$

The final expression for the wing torsion box weight (4.2.1) taking intoaccount (4.2.3) and (4.2.16) takes on the form:

$\begin{matrix}{G_{w} = {2 \cdot \gamma \cdot l \cdot \left\{ {\frac{M}{h_{ef} \cdot \sigma} + {{\lambda \cdot {\frac{B^{1/2} \cdot \sigma^{2}}{\left\lbrack {\delta \cdot \left( {\frac{M}{h_{ef} \cdot B} - {\delta \cdot \sigma}} \right) \cdot \eta_{\delta}^{1/4} \cdot E^{7}} \right\rbrack^{1/4}}++}}{h_{ef} \cdot K_{{ef}.w} \cdot \left( {K_{\tau.w} \cdot E \cdot \sqrt{\eta_{\tau.w}}} \right)^{{- 1}/5}}{\quad{\left\lbrack {\frac{2}{3} \cdot F_{pi}^{2} \cdot \left. \quad\left( {\frac{M_{ts}}{2 \cdot K_{ts} \cdot \Omega} + \frac{Q_{{tr}.d}}{K_{{ef}.w} \cdot h_{w}}} \right) \right\rbrack^{1/5}} \right\}.}}}} \right.}} & \left( {4.2{.17}} \right)\end{matrix}$

Analysis of (4.2.17) shows the wing torsion box weight to depend onthree variable parameters, the panel stress σ, the skin thickness δ andthe effective wing height h_(ef) possessing an optimum in each of these.The pillar area F_(pi) is selected by analogy with the existingstructures.

Combination of the three variable parameters σ, δ and h_(ef)corresponding to the torsion box minimum weight calculated through(4.2.17) are the sought-for optimal parameters. To determine these, onecan employ the scanning method.

The optimal in weight values of the torsion box member cross-sections,viz. the dimensions of stringer and rib shapes, distances betweenstringers and ribs, skin thickness, web thickness, distance betweenpillars, can be easily calculated, as these are functions of the foundoptimal parameters σ, δ and h_(ef).

4.3 Formulae for Calculation of Optimal Parameters in Case of JointAction of Normal and Transversal Loads in the Wing Torsion Box Members

In this section, the formulae are derived necessary for numericalsolution of the formulated problem.

The weight of the wing torsion box is equal to the sum of weight of itsload-bearing members:

G=2·γ·(F _(s) ·n _(s) ·l+δ·B·l+F _(r) ·n _(r) ·B+F _(pi) ·n _(pi) ·h_(w))  (4.3.1),

n_(s), n_(r) and n_(pi) being, respectively, number of stringers, ribsand pillars in the torsion box:n_(s)=B/b_(s), n_(r)=l/l_(r), n_(pi)=l/r.

It is obvious from (4.3.1) that the weight of the torsion box depends oneight parameters, namely F_(s), b_(s), δ, F_(r), l_(r), F_(pi), r,h_(w). In their turn, areas of the shapes F_(s), F_(r), F_(pi) arefunctions of load-carrying members shape dimensions.

The geometrical characteristics of the stringer shape could bedetermined analytically given the force acting onto the stringer.

The force acting onto the stringer can be written as

$\begin{matrix}{P_{s} = {b_{s} \cdot {\left( {\frac{F_{p} \cdot \sigma}{B} - {\delta \cdot \sigma}} \right).}}} & \left( {4.3{.2}} \right)\end{matrix}$

To obtain the panel cross-section area F_(p) in (4.3.2), let us employthe equation for the wing bending strength.

$\begin{matrix}{\frac{M}{W} \leq {\sigma.}} & \left( {4.3{.3}} \right)\end{matrix}$

The resistance torque of the wing torsion box equals to

$\begin{matrix}{W = {{F_{p} \cdot h_{ef}} + {2 \cdot {\frac{t \cdot h_{w}^{2}}{6}.}}}} & \left( {4.3{.4}} \right)\end{matrix}$

Hence, the required wing torsion box panel area providing for the givenbending torque is

$\begin{matrix}{F_{p} = {\frac{W}{h_{ef}} - {\frac{t \cdot h_{w}^{2}}{3 \cdot h_{ef}}.}}} & \left( {4.3{.5}} \right)\end{matrix}$

Taking into account (4.3.3), the required panel area is

$\begin{matrix}{F_{p} = {\frac{M}{\sigma \cdot h_{ef}} - {\frac{t \cdot h_{w}^{2}}{3 \cdot h_{ef}}.}}} & \left( {4.3{.6}} \right)\end{matrix}$

It follows from (4.3.6) that taking into account operation of webs inthe course of the wing bending enables reduction of panel areas.

The expression (4.3.2), taking into account (4.3.6), takes on the form:

$\begin{matrix}{P_{c} = {b_{c} \cdot {\left( {\frac{M}{h_{ef} \cdot B} - \frac{\sigma \cdot t \cdot h_{w}^{2}}{3 \cdot h_{ef} \cdot B} - {\delta \cdot \sigma}} \right).}}} & \left( {4.3{.7}} \right)\end{matrix}$

The first term in parentheses of (4.3.7) is the linear force acting ontothe panel, the second one accounts for reducing of the linear forceacting onto the panel due to operation of webs at bending, and the thirdone accounts for reduction of the linear force due to operation of skinat bending.

The distance between stringers b_(s) in the expression (4.3.7) isdetermined by the known formulae, see (2.2.2) and (2.5.2).

Width of the skin cell b_(sk) (the distance between riveted joints ofstringer fastening to the skin) in expressions (2.2.2) and (2.5.2) isderived from the condition of the skin cell strength at joint operationof normal and tangential stresses.

The condition of the skin cell strength has the form

$\begin{matrix}{{{\frac{\sigma}{\sigma_{{cr} \cdot {sk}}} + \left( \frac{\tau}{\tau_{{cr} \cdot {sk}}} \right)^{2}} = 1},} & \left( {4.3{.8}} \right)\end{matrix}$

σ_(cr.sk) being the normal critical strength in the skin cell,

$\begin{matrix}{{\sigma_{{cr} \cdot {sk}} = {\frac{K_{\sigma \cdot {sk}} \cdot E}{\left( {b_{sk}/\delta} \right)^{2}} \cdot \sqrt{\eta_{\sigma \cdot {sk}}}}},} & \left( {4.3{.9}} \right)\end{matrix}$

τ_(cr.sk) being the tangential critical strength in the skin cell,

$\begin{matrix}{{\tau_{{cr} \cdot {sk}} = {\frac{K_{\tau \cdot {sk}} \cdot E}{\left( {b_{sk}/\delta} \right)^{2}} \cdot \sqrt{\eta_{\tau \cdot {sk}}}}},} & \left( {4.3{.10}} \right)\end{matrix}$

τ_(sk) being the tangential stress in the skin.

$\begin{matrix}{\tau_{sk} = {\frac{M_{ts}}{2 \cdot K_{ts} \cdot \Omega \cdot t}.}} & \left( {4.3{.11}} \right)\end{matrix}$

The tangential stresses in the skin brought about by the transversalforces are relatively small and are therefore neglected in (4.3.11).

The factors K_(σ.sk) and K_(τ.sk) are determined for the pivotedinfinitely long plate: K_(σ.sk)=3.6, K_(τ.sk)=4.8.

Ductility factors for compression η_(σ.sk) and for shear η_(τ.sk) arederived, respectively, from equations (1.2.5) and (4.2.7).

Solving the equation (4.3.8) for b_(sk) taking into account (4.3.9) and(4.3.10), we get the biquadratic equation

$\begin{matrix}{{{{x \cdot b_{sk}^{4}} + {y \cdot b_{sk}^{2}} - 1} = 0},{where}} & \left( {4.3{.12}} \right) \\{{x = \frac{\tau_{sk}^{2}}{K_{\tau}^{2} \cdot E^{2} \cdot \delta^{4} \cdot \eta_{\tau \cdot {sk}}}},} & \left( {4.3{.13}} \right) \\{y = {\frac{\sigma}{K_{\sigma \cdot {sk}} \cdot E \cdot \delta^{2} \cdot \sqrt{\eta_{\sigma \cdot {sk}}}}.}} & \left( {4.3{.14}} \right)\end{matrix}$

The solution of the equation (4.3.12) is

$\begin{matrix}{b_{sk} = {\left\lbrack {\sqrt{\left( \frac{y}{2 \cdot x} \right)^{2} + \frac{1}{x}} - \frac{y}{2x}} \right\rbrack^{\frac{1}{2}}.}} & \left( {4.3{.15}} \right)\end{matrix}$

The distance r between web-reinforcing pillars in (4.3.1) is determinedfrom the condition of web cell strength at joint operation of normal andtangential stresses.

The web cell strength condition has the form

$\begin{matrix}{{{\left( \frac{\sigma_{w}}{\sigma_{{cr} \cdot w}} \right)^{2} + \left( \frac{\tau_{w}}{\tau_{{cr} \cdot w}} \right)^{2}} = 1},} & \left( {4.3{.16}} \right)\end{matrix}$

σ_(cr.w) being the normal critical stress in the web cell,

$\begin{matrix}{{\sigma_{{cr} \cdot w} = {\frac{K_{\sigma \cdot w} \cdot E}{\left( {r/t} \right)^{2}} \cdot \sqrt{\eta_{\sigma \cdot w}}}},} & \left( {4.3{.17}} \right)\end{matrix}$

τ_(cr.w) the tangential critical stress in the web cell,

$\begin{matrix}{{\tau_{{cr} \cdot w} = {\frac{K_{\tau \cdot w} \cdot E}{\left( {r/t} \right)^{2}} \cdot \sqrt{\eta_{\tau \cdot w}}}},} & \left( {4.3{.18}} \right)\end{matrix}$

σ_(w) the normal stress in the web,

$\begin{matrix}{{\sigma_{w} = {\sigma \cdot \frac{h_{ef}}{h_{w}}}},} & \left( {4.3{.19}} \right)\end{matrix}$

τ_(w) the tangential stress in the web determined from equation (4.2.9).The factors K_(σ.w) and K_(τ.w) are determined for the pivoted plate andare depending on the web cell overall dimensions in compliance the Table4.1.Ductility factors at compression η_(σ.w) and at shear η_(τ.w) aredetermined, respectively, according to equations (1.2.5) and (4.2.7).The equation (4.3.16) is solved numerically, as the factors K_(σ.w) andK_(τ.w) depend on the sought-for r (Table 4.1, FIG. 75).

The pillar area F_(pi) in (4.3.1) is derived by the method outlined insection 3.2 from the expression

$\begin{matrix}{{F_{pi} = {{\Psi_{pi}^{2} \cdot F_{pi}^{\prime}} = {\left( \frac{J_{pi}}{J_{pi}^{\prime}} \right)^{1/2} \cdot F_{pi}^{\prime}}}},} & \left( {4.3{.20}} \right)\end{matrix}$

J_(pi)′ and F_(pi)′ being, respectively, the moment of inertia and thearea of the pillar shape selected basing on the analysis of the existingdesigns or on constructive considerations.The pillar moment of inertia J_(pi) appearing in (4.3.20) shall beselected so that the critical stress of the pillar-reinforced web (atthe length l) and the web cell critical stress (at the length r) beequal. Increasing the moment of inertia in excess of this value issenseless, as no increase in the critical stress would be therebyreached.

The pillar moment of inertia in this case shall be:

$\begin{matrix}{{J_{pi} = {{\theta \cdot \frac{t^{3}}{12 \cdot \left( {1 - \mu^{2}} \right)} \cdot l} = \frac{\theta \cdot t^{3} \cdot l}{10.7}}},} & \left( {4.3{.21}} \right)\end{matrix}$

θ being the factor depending on ratios of the web overall dimensions andthe number of pillars;μ is the Poisson factor, μ=0.31.With increasing the number of pillar, the factor θ increases. Assuming,that in all practical cases the required pillar moment of inertia shalldiffer from that for the plate with one rib in the middle not more thanby factor of two, we come to results summarized in the Table 4.2, FIG.76.Taking into account that

${\frac{l}{h_{w}} > 2},$

we assume, providing for the strength margin θ=1.66.For the selected value of θ, the equation (4.3.21) holds with thestrength margin for any pitch of the equidistant pillars.

The equation for θ provided a high number of pillars has the form

$\begin{matrix}{{\theta = \frac{2{\sum\; {{B_{i} \cdot \sin}\frac{\pi \cdot c_{i}}{r}}}}{D \cdot h_{w}}},} & \left( {4.3{.22}} \right)\end{matrix}$

where B_(i) is the bending rigidity of the pillar located from the webedge at the distance c_(i);

$D = \frac{E \cdot t^{3}}{12 \cdot \left( {1 - \mu^{2}} \right)}$

is the web cylindrical rigidity.

In the case r=h_(w), it follows from (4.3.22) that θ=0; that is, thepillar shape moment of inertia, hence its area (4.3.20) are equal tozero. Provided r≠h, the equation (4.3.20) can be employed to refine thevalue of the factor θ assumed providing the strength margin.

Analysis of the parameters appearing in the equation for the torsion boxweight has shown that the following four parameters shall be assignedthe variables: h_(ef), δ, σ, and t.

Algorithm for Determining the Optimal Parameters for Wing Torsion BoxMembers

This algorithm takes into account the joint loading of the panel and thewebs with normal and tangential stresses (section 4.3), joint operationof the skin with stringers and ribs at bending (section 3.3), theoptimal inclination angle of the web for the U-shaped stringer (section2.5), and the shape dimensions of the web-reinforcing pillar aredetermined (section 4.3).

(1) The benchmark data:

-   -   loads in the wing design section: bending torque M, torsion        torque M_(ts), transversal force Q;    -   distance between design wing sections l;    -   aerodynamic profile of the wing with the given torsion box        dimensions; it is assumed that h_(w1)=h_(w2)=h_(ef), Ω=h_(ef)·B;    -   K_(ef), the factor relating the effective height of the wing        torsion box with the maximum height of the aerodynamic profile;        it is assumed that K_(ef)=1;    -   K_(w) the factor relating vertical web height with the maximum        height of aerodynamic profile; it is assumed that K_(w)=1;    -   K_(ts), the factor relating the given torsion box outline area        with the design one; it is assumed that K_(ts)=1;    -   the optimal stringer shape coefficients accounting for        technological and design requirements a₁, a₂, a₃, a₄, A, A₁, A₂,        A₃, K_(s) (Table 4.3, FIG. 77);    -   the benchmark dimensions and geometrical characteristics of the        rib shape b′_(r), δ′_(b.r), C′_(r), δ′_(c.r), F′_(r), W′_(r),        J′_(r), f_(r) (Table 4.4, FIG. 78);    -   the rib rigidity factor K_(ex); it is assumed that K_(ex)=1        (section 3.3);    -   the factor accounting for the condition of the stringer leaning        on the rib c; it is assumed that c=2 (section 3.3);    -   the coefficients in the equations for critical stresses of skin        and web accounting for cell dimensions, leaning conditions and        type of loading K_(σ.sk)=3.6, K_(τ.sk)=4.8, K_(σ.w), K_(τ.w),        (Table 4.1, FIG. 75);    -   the benchmark dimensions and geometrical characteristics of        pillar shape b′_(pi), δ′_(b.pi), c′_(pi), δ′_(c.pi), F′_(pi),        J′_(pi), W′_(r). (Table 4.5, FIG. 79);    -   the coefficient in the equation for determining of the pillar        shape moment of inertia θ=1.66;    -   the factor depending on the number of spans in the equation for        rib rigidity β=0.255 (section 3.1);    -   angle of web obliquity; it is assumed that α_(w)=0.

(2) The iteration cycles are specified in the following variableparameters:

-   -   the effective wing profile height h_(ef); e.g., the range of its        variation is assumed to be h_(ef)=(50-200) cm;    -   skin thickness δ; the series of values in compliance with the        existing standards is adopted, for example, δ=0.1; 0.12; 0.15;        0.2; 0.25; 0.3; 0.35; 0.4; 0.45; 0.5; 0.6; 0.8; 1.0 [cm];    -   web thickness t; the series of values in compliance with the        existing standards is adopted, for example, t=0.1; 0.12; 0.15;        0.2; 0.25; 0.3; 0.35; 0.4; 0.45; 0.5; 0.6; 0.8; 1.0 [cm];    -   panel normal stress σ; the range of variation is adopted        σ=(500−σ_(0.2))[daN/cm2].

In specifying cycles in h_(ef) and σ, increments of the variedparameters, Δh_(ef) and Δσ, are reduced until the subsequent value forthe weight function would differ from the previous one by no more than2%.

The height of vertical webs is calculated

$h_{w} = {\frac{K_{w}}{K_{ef}} \cdot {h_{ef}.}}$

The area of the torsion box reacting the torsion torque:

Ω_(d) =K _(ts)·Ω

The tangential stress in the skin is determined

$\tau_{sk} = {\frac{M_{ts}}{2 \cdot \Omega_{d} \cdot t}.}$

The ductility factor for normal stresses in the panel is calculated

$\eta_{\sigma} = \left\{ {\begin{matrix}{1,} & {{{provided}\mspace{14mu} \sigma} \leq \sigma_{pr}} \\{{\frac{\sigma}{\sigma_{pr}}\left( \frac{\sigma_{o} - \sigma}{\sigma_{o} - \sigma_{pr}} \right)^{2}},} & {{{provided}\mspace{14mu} \sigma} > \sigma_{pr}}\end{matrix}.} \right.$

The value of y and the value of x at η_(τ.sk)=1 is calculated

${y = \frac{\sigma}{K_{\sigma \cdot {sk}} \cdot E \cdot \delta^{2} \cdot \sqrt{\eta_{\sigma}}}},{x_{(\eta_{{\tau \cdot {sk}} = 1})} = {\frac{\tau_{sk}^{2}}{K_{\tau \cdot {sk}}^{2} \cdot E^{2} \cdot \delta^{4}}.}}$

The width of the skin cell at η_(τ.sk)=1 is calculated

$b_{{sk}{({\eta_{\tau \cdot {sk}} = 1})}} = \left\lbrack {\sqrt{\left( \frac{y}{2 \cdot x_{({\eta_{\tau \cdot {sk}} = 1})}} \right)^{2} + \frac{1}{x_{({\eta_{\tau \cdot {sk}} = 1})}}} - \frac{y}{2 \cdot x_{({\eta_{\sigma \cdot {sk}} = 1})}}} \right\rbrack^{\frac{1}{2}}$

The critical tangential stress in the skin cell at η_(τ.sk)=1 iscalculated

$\tau_{{cr} \cdot {{sk}{({\eta_{\tau \cdot {sk}} = 1})}}} = {\frac{K_{\tau \cdot {sk}} \cdot E}{\left( \frac{b_{{sk}{({\eta_{\tau \cdot {sk}} = 1})}}}{\delta} \right)^{2}}.}$

The shear ductility factor for the skin is calculated. If τ_(cr.sk(η)_(τ.sk) ₌₁₎≤τ_(pr), then η_(τ.sk)=1, otherwise

$\eta_{\tau \cdot {sk}} = {\left\{ \frac{\left\lbrack {\sqrt{{\tau_{pr} \cdot \left( {\tau_{0} - \tau_{pr}} \right)^{2}} + {4 \cdot \tau_{{cr} \cdot {{sk}{({\eta_{\tau \cdot {sk}} = 1})}}}^{2} \cdot \tau_{0}}} - {\tau_{pr}^{\frac{1}{2}} \cdot \left( {\tau_{0} - \tau_{pr}} \right)}} \right\rbrack^{2}}{4 \cdot \tau_{{cr} \cdot {{sk}{({\eta_{\tau \cdot {sk}} = 1})}}}^{3}} \right\}^{2}.}$

The critical tangential stress in the skin cell is calculated

τ_(cr.sk)=τ_(cr.sk(η) _(τ.sk) ₌₁₎·η_(τ.sk) ^(1/2).

The width of skin cell is calculated

$b_{sk} = {\left\lbrack {\sqrt{\left( \frac{y \cdot \eta_{\tau \cdot {sk}}}{2 \cdot x_{({\eta_{\tau \cdot {sk}} = 1})}} \right)^{2} + \frac{\eta_{\tau \cdot {sk}}}{x_{({\eta_{\tau \cdot {sk}} = 1})}}} - \frac{y \cdot \eta_{\tau \cdot {sk}}}{2 \cdot x_{({\eta_{\tau \cdot {sk}} = 1})}}} \right\rbrack^{\frac{1}{2}}.}$

The pitch of stringer installation is determined:

-   -   for shapes forming with the sheet an open configuration        b_(s)=2·b_(re),    -   for those forming with the sheet a close configuration        b_(s)=b_(re).

The number of stringers is determined

$n_{s} = {\frac{B}{b_{s}}.}$

The value of n_(s) is rounded to the nearest higher integer.

The distance between stringers is refined

$b_{s} = {\frac{B}{n_{s}}.}$

The force per one stringer is determined

$P_{s} = {b_{s} \cdot {\left( {\frac{M}{h_{ef} \cdot B} - \frac{\sigma \cdot t \cdot h_{w}^{2}}{3 \cdot B \cdot h_{ef}} - {\delta \cdot \sigma}} \right).}}$

The distance between ribs is determined

$l_{r} = {\left\lbrack \frac{\pi^{4} \cdot E^{3} \cdot c^{2} \cdot P_{s}^{2} \cdot \left( {\sqrt{\eta_{\sigma}} \cdot A} \right)^{5}}{\sigma^{5}} \right\rbrack^{\frac{1}{4}}.}$

The number of ribs is calculated

$n_{r} = {\frac{l}{l_{r}}.}$

The value of n_(r) is rounded to the nearest higher integer.

The distance between ribs is refined

$l_{r} = {\frac{l}{n_{r}}.}$

The characteristic stringer shape dimensions are determined

${a = {\left( \frac{P_{s} \cdot l_{r}}{\pi \cdot \sigma \cdot c^{1/2}} \right)^{1/3} \cdot \frac{A_{1}}{\eta_{\sigma}^{1/12}}}},{\delta_{a} = {\left( \frac{\pi \cdot P_{s}^{2} \cdot c^{\frac{1}{2}}}{\sigma^{2} \cdot l_{r}} \right)^{1/3} \cdot A_{2} \cdot {\eta_{\sigma}^{1/12}.}}}$

The other dimensions of the stringer shape are calculated

b=a·a ₂, δ_(b)=δ_(a) ·a ₄,

c=b·a ₁, δ_(c)=δ_(b) ·a ₃.

For the U-shaped stringer, the lateral web inclination angle isdetermined

$\alpha = {{arc}\mspace{11mu} {{\sin\left( \frac{b_{sk} - a - c}{2 \cdot b} \right)}.}}$

If α≥60°, then it is assumed that α=60°; if α<0°, then it is assumedthat α=0°. Branch to step in paragraph [000698].

The distance between stringers for 0<α≤60° is calculated:

b_(s)=c+2·b·sin α+a+b_(sk). Then steps in paragraphs [000687] to[000694] are reiterated, and after the step in paragraph [000694] branchto the step in paragraph [000698].

The stringer shape area is determined:

for shapes forming with the sheet a closed configuration (U-shapedprofile):

F _(s) =a·δ _(a)·(1+2·a ₂ ·a ₄+2·a ₁ ·a ₂ ·a ₃ ·a ₄)

for shapes forming with the sheet an open configuration:

(Z-shaped profile) F_(s)=a·δ_(a)·(1+2·a₁·a₂),(L-shaped profile) F_(s)=a·δ_(a)·(1+a₁·a₂).

The reduced stringer shape area is determined:

F _(re.s) =F _(s) +b _(c)·δ.

The stringer shape moment of inertia is determined:

J _(s) =a ³·δ_(a) ·A ₄·cos² α.

The coordinate of the stringer shape neutral axis is determined:

For U-shaped profile:

${y_{s} = \frac{a \cdot \left( {1 + {a_{2} \cdot a_{4}}} \right)}{1 + {2 \cdot \left( {{a_{2} \cdot a_{4}} + {a_{1} \cdot a_{2} \cdot a_{3} \cdot a_{4}}} \right)}}},$

For Z-shaped profile:

${y_{s} = \frac{a}{2}},$

For L-shaped profile:

$y_{s} = {\frac{a}{2 \cdot \left( {1 + {a_{1} \cdot a_{2}}} \right)}.}$

The coordinate of the stringer shape neutral axis is determined for thestringer with the reduced skin:

$y_{{re} \cdot s} = {\frac{F_{s}}{F_{{re} \cdot s}} \cdot {y_{s}.}}$

The reduced stringer shape moment of inertia is determined

J _(re.s) =J _(s) +F _(s)(y _(s) −y _(re.s))² +b _(s) ·δ·y _(re.s).

The required rib shape moment of inertia is determined:

$J_{r \cdot p} = {\frac{c \cdot \pi^{2} \cdot J_{r \cdot {rq}} \cdot B^{3} \cdot \eta_{\sigma}}{48 \cdot K_{ex} \cdot \beta \cdot l_{r}^{3}}.}$

The coordinate of the rib shape neutral axis with reduced skin isdetermined:

$Y_{r \cdot {re}} = {\frac{F_{r}^{\prime} \cdot b_{r}^{\prime}}{2 \cdot \left( {F_{r}^{\prime} + {b_{re} \cdot \delta}} \right)}.}$

Rib shape moment of inertia is determined:

J _(r) =J _(r.rq) −b _(sk) ·δ·Y ² _(r.re).

The rib shape area is determined

$F_{r} = {\left( \frac{J_{r}}{f_{r}} \right)^{1/2}.}$

The pillar moment of inertia is determined

$J_{pi} = {\frac{\theta \cdot t^{3} \cdot l}{10.7}.}$

The pillar shape area is determined

$F_{pi} = {F_{pi}^{\prime} \cdot {\left( \frac{J_{pi}}{J_{pi}^{\prime}} \right)^{1/2}.}}$

r is assigned a number of values, for example: r=h_(ef); 0.8 h_(ef); 0.6h_(ef); 0.4 h_(ef); 0.2 h_(ef).

The critical normal stress in the web cell is determined at η_(σ.w)=1,

$\sigma_{{cr} \cdot {w{({\eta_{\sigma \cdot w} = 1})}}} = {\frac{K_{\sigma \cdot w} \cdot E}{\left( {r/t} \right)^{2}}.}$

The compression ductility factor for the web is calculated. Ifσ_(cr.w(η) _(σ) _(.w=1))≤σ_(pr), then η_(σ.w)=1, otherwise

$\eta_{\sigma \cdot w} = {\left\{ \frac{\left\lbrack {\sqrt{{\sigma_{pr} \cdot \left( {\sigma_{0} - \sigma_{pr}} \right)^{2}} + {4 \cdot \sigma_{{cr} \cdot {w{({\eta_{\sigma \cdot w} = 1})}}}^{2} \cdot \sigma_{0}}} - {\sigma_{pr}^{1/2} \cdot \left( {\sigma_{0} - \sigma_{pr}} \right)}} \right\rbrack^{2}}{4 \cdot \sigma_{{cr} \cdot {w{({\eta_{\sigma \cdot w} = 1})}}}^{3}} \right\}^{2}.}$

The critical normal stress in the web cell is determined

σ_(cr.w)=σ_(cr.w(η) _(σ.w) ₌₁₎·η_(σ.w) ^(1/2).

The transversal force in the web is determined

$Q_{{tr} \cdot d} = {\frac{1}{2} \cdot {\left( {Q - {\frac{M_{ts}}{h_{w}} \cdot \alpha_{w}}} \right).}}$

The tangential stress in the web is determined

$\tau_{w} = {\frac{M_{ts}}{2{\Omega_{d} \cdot t}} + {\frac{Q_{{tr} \cdot d}}{h_{w} \cdot t}.}}$

The critical tangential stress in the web is determined at η_(τ.w)=1

$\tau_{{cr} \cdot {w{({\eta_{\tau \cdot w} = 1})}}} = {\frac{K_{\tau \cdot w} \cdot E}{\left( {r/t} \right)^{2}}.}$

The shear ductility factor of the web is determined. If τ_(cr.w(η)_(τ.w) ₌₁₎≤τ_(pr), then η_(τ.w)=1, otherwise

$\eta_{\tau \cdot w} = {\left\{ \frac{\left\lbrack {\sqrt{{\tau_{pr} \cdot \left( {\tau_{0} - \tau_{pr}} \right)^{2}} + {{4 \cdot \tau_{{cr} \cdot {w{({\eta_{\tau \cdot w} = 1})}}}^{2}}\tau_{0}}} - {\tau_{pr}^{\frac{1}{2}} \cdot \left( {\tau_{0} - \tau_{pr}} \right)}} \right\rbrack^{2}}{4 \cdot \tau_{{cr} \cdot {w{({\eta_{\tau \cdot w} = 1})}}}^{3}} \right\}^{2}.}$

The critical tangential stress in the web is determined

τ_(cr.w)=τ_(cr.w(η) _(τ.w) ₌₁₎·η_(τ.w) ^(1/2)

The normal stress in the web is determined

$\sigma_{w} = {\sigma \cdot {\frac{h_{w}}{h_{ef}}.}}$

The distance between pillars is selected basing on the web strengthcondition. If

${\left( \frac{\sigma_{w}}{\sigma_{{cr} \cdot w}} \right)^{2} + \left( \frac{\tau_{w}}{\tau_{{cr} \cdot w}} \right)^{2}} > 1$

then branch to step in paragraph [000709].

The number of pillars is determined

$n_{pi} = {\frac{l}{r}.}$

The value of n_(pi) is rounded to the nearest higher integer.

The distance between pillars is refined

$r = {\frac{l}{n_{pi}}.}$

The variable parameters h_(ef), t, δ and σ are determined basing oncondition of the torsion box minimum weight

G=2·γ·(F _(s) ·n _(s) l+δ·B·l+F _(r) ·n _(r) ·B+h _(w) ·t·l+F _(pi) ·n_(pi) ·h _(w)).

The factor of the rib profile dimension change is calculated

$\Psi_{r} = {\left( \frac{J_{r}}{J_{r}^{\prime}} \right)^{\frac{1}{4}}.}$

The optimal rib shape dimensions are calculated

b _(r)=Ψ_(r′) ·b′ _(r) , c _(r)=Ψ_(r) ·c′ _(r),

δ_(b.r)=Ψ_(r)·δ_(b.r), δ_(c.r)=Ψ_(r)·δ′_(c.r).

The factor of the pillar shape dimension change is calculated

$\Psi_{pi} = {\left( \frac{J_{pi}}{J_{pi}^{\prime}} \right)^{\frac{1}{4}}.}$

The optimal pillar shape dimensions are calculated

b _(pi)=Ψ_(pi) ·b′ _(pi) , c _(pi)=Ψ_(pi) ·c′ _(pi),

δ_(c.pi)=⋅_(pi)·δ′_(c.pi), δ_(c.pi)=Ψ_(pi)·δ′_(c.pi).

The radius of inertia of the stringer shape with reduced skin iscalculated

$i_{{re},s} = {\left( \frac{J_{{re} \cdot s}}{F_{{re} \cdot s}} \right)^{\frac{1}{2}}.}$

The stringer overall stability critical stress for η_(σ.tot)=1 iscalculated.

$\sigma_{\kappa \; {p \cdot {{tot}{({\eta_{\sigma \cdot {tot}} = 1})}}}} = \frac{c \cdot \pi^{2} \cdot E}{\left( {l/i_{{re} \cdot s}} \right)^{2}}$

The ductility factor of the stringer overall stability critical stressis calculated. If σ_(cr.tot(η) _(σ.tot) ₌₁₎≤τ_(pr), then η_(σ.tot)=1,otherwise

$\eta_{\sigma \cdot {tot}} = {\frac{\sigma_{0}}{\sigma_{{cr} \cdot {{tot}{({\eta_{\sigma \cdot {tot}} = 1})}}}} - {\frac{\sigma_{0} - \sigma_{pr}}{\sigma_{{cr} \cdot {{tot}{({\eta_{\sigma \cdot {tot}} = 1})}}}} \cdot {\left( \frac{\sigma_{pr}}{\sigma_{{cr} \cdot {{tot}{({\eta_{\sigma \cdot {tot}} = 1})}}}} \right)^{1/2}.}}}$

The stringer overall stability critical stress is calculated.

σ_(cr.tot)=σ_(cr.tot(η) _(σ.tot) ₌₁₎·η_(σ.tot)

The stringer local stability normal critical stress for η_(σ.l)=I iscalculated.

$\sigma_{{cr} \cdot {l{({\eta_{\sigma \cdot l} = 1})}}} = {\frac{K_{s} \cdot E}{\left( {a/\delta_{a}} \right)^{2}}.}$

The ductility factor of the stringer local stability critical stress iscalculated. If σ_(cr.l(η) _(σ.l) ₌₁₎≤σ_(pr) then η_(σ.l)=1, otherwise

$\eta_{\sigma \cdot l} = {\left\{ \frac{\left\lbrack {\sqrt{{\sigma_{pr} \cdot \left( {\sigma_{0} - \sigma_{pr}} \right)^{2}} + {4 \cdot \sigma_{{cr} \cdot {({\eta_{\sigma \cdot l} = 1})}}^{2} \cdot \sigma_{0}}} - {\sigma_{pr}^{\frac{1}{2}} \cdot \left( {\sigma_{0} - \sigma_{pr}} \right)}} \right\rbrack^{2}}{4 \cdot \sigma_{{cr} \cdot {l{({\eta_{\sigma \cdot l} = 1})}}}^{3}} \right\}^{2}.}$

The stringer local stability critical stress is calculated.

σ_(cr.l)=σ_(cr.l(η) _(σ.l) ₌₁₎=η_(σ.l) ^(1/2).

The skin normal critical stress is calculated.

$\sigma_{{cr} \cdot l} = \frac{K_{\sigma \cdot {sk}} \cdot E}{\left( {b_{sk}/\delta} \right)^{2}}$

Output are the following wing torsion box optimal parameters:

-   -   shape dimensions of stringer, rib, pillar: a, δ_(a), b, δb, c,        δ_(c), b_(r), δ_(b.r), c_(r), b_(pi), δ_(b.pi), c_(pi),        δ_(c.pi);    -   distances between stringers, ribs and pillars b_(s), l_(r), r;    -   values of thickness for web and skin t and δ;    -   effective wing height h_(ef);    -   normal and tangential stresses of stringer, skin and web σ,        τ_(sk), τ_(w);    -   critical normal and tangential stresses for stringer, skin cell        and web cell σ_(cr.l), σ_(cr.tot), σ_(cr.sk), σ_(cr.w),        τ_(cr.sk), τ_(cr.w);    -   inclination angle α for the U-shaped stringer lateral web.

INDUSTRIAL APPLICABILITY

TPM and TPM-panels as contemplated herein may be of any known grade ofstructural metallic alloys and/or non-metal material so long as suitableto satisfy desired constructive restrictions. The proposed techniquesfor improved efficiency in designing TPM and TPM-panel configurationsfor industrial applications, with the intent of their being fitted intoa further manufactured profile or system design and taking any feasibleto design shape, including fully monolithic TPM-panels, and sandwichTPM-panels with longitudinal load-bearing filler, corrugated componentsand other shapes depending on specific conditions.

It is believed that TPM and TPM-panels which utilize the proposedmodeling and analysis tool will often result in a reduction of structureweight improvement of between 5 and 35 percent or more, as mightotherwise be feasible using conventional tools.

It will be understood that each of the elements described above, or twoor more together, may also find a useful application in other types ofmethods differing from the type described above.

While the invention has been illustrated and described as embodied in amethod of producing thin wall profile members, it is not intended to belimited to the details shown, since various modifications and structuralchanges may be made without departing in any way from the spirit of thepresent invention.

Without further analysis, the foregoing will so fully reveal the gist ofthe present invention that others can, by applying current knowledge,readily adapt it for various applications without omitting featuresthat, from the standpoint of prior art, fairly constitute essentialcharacteristics of the generic or specific aspects of this invention.

Those of skill would further appreciate that the various illustrativelogical blocks, modules, circuits, and algorithm steps described inconnection with the embodiments disclosed herein may be implemented aselectronic hardware, computer software, or combinations of both. Toclearly illustrate this interchangeability of hardware and software,various illustrative components, blocks, modules, circuits, and stepshave been described above generally in terms of their functionality.Whether such functionality is implemented as hardware or softwaredepends upon the particular application and design constraints imposedon the overall system. Skilled artisans may implement the describedfunctionality in varying ways for each particular application, but suchimplementation decisions should not be interpreted as causing adeparture from the scope of the exemplary embodiments of the invention.

The various illustrative logical blocks, modules, and circuits describedin connection with the embodiments disclosed herein may be implementedor performed with a general purpose processor, a Digital SignalProcessor (DSP), an Application Specific Integrated Circuit (ASIC), aField Programmable Gate Array (FPGA) or other programmable logic device,discrete gate or transistor logic, discrete hardware components, or anycombination thereof designed to perform the functions described herein.A general purpose processor may be a microprocessor, but in thealternative, the processor may be any conventional processor,controller, microcontroller, or state machine. A processor may also beimplemented as a combination of computing devices, e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration.

The steps of a method or algorithm described in connection with theembodiments disclosed herein may be embodied directly in hardware, in asoftware module executed by a processor, or in a combination of the two.A software module may reside in Random Access Memory (RAM), flashmemory, Read Only Memory (ROM), Electrically Programmable ROM (EPROM),Electrically Erasable Programmable ROM (EEPROM), registers, hard disk, aremovable disk, a CD-ROM, or any other form of storage medium known inthe art. An exemplary storage medium is coupled to the processor suchthat the processor can read information from, and write information to,the storage medium. In the alternative, the storage medium may beintegral to the processor. The processor and the storage medium mayreside in an ASIC. The ASIC may reside in a user terminal. In thealternative, the processor and the storage medium may reside as discretecomponents in a user terminal.

In one or more exemplary embodiments, the functions described may beimplemented in hardware, software, firmware, or any combination thereof.If implemented in software, the functions may be stored on ortransmitted over as one or more instructions or code on acomputer-readable medium. Computer-readable media includes both computerstorage media and communication media including any medium thatfacilitates transfer of a computer program from one place to another. Astorage media may be any available media that can be accessed by acomputer. By way of example, and not limitation, such computer-readablemedia can comprise RAM, ROM, EEPROM, CD-ROM or other optical diskstorage, magnetic disk storage or other magnetic storage devices, or anyother medium that can be used to carry or store desired program code inthe form of instructions or data structures and that can be accessed bya computer. Also, any connection is properly termed a computer-readablemedium. For example, if the software is transmitted from a website,server, or other remote source using a coaxial cable, fiber optic cable,twisted pair, digital subscriber line (DSL), or wireless technologiessuch as infrared, radio, and microwave, then the coaxial cable, fiberoptic cable, twisted pair, DSL, or wireless technologies such asinfrared, radio, and microwave are included in the definition of medium.Disk and disc, as used herein, includes compact disc (CD), laser disc,optical disc, digital versatile disc (DVD), floppy disk and blu-ray discwhere disks usually reproduce data magnetically, while discs reproducedata optically with lasers. Combinations of the above should also beincluded within the scope of computer-readable media.

The previous description of the disclosed exemplary embodiments isprovided to enable any person skilled in the art to make or use thepresent invention. Various modifications to these exemplary embodimentswill be readily apparent to those skilled in the art, and the genericprinciples defined herein may be applied to other embodiments withoutdeparting from the spirit or scope of the invention. Thus, the presentinvention is not intended to be limited to the embodiments shown hereinbut is to be accorded the widest scope consistent with the principlesand novel features disclosed herein.

I claim:
 1. A thin wall profile member (TPM) with a cross-sectionalshape and dimensions constant along its length, the TPM beingpre-constructed for use as a load carrying, weight optimized structuralelement under compressive load, the TPM comprising: at least one mainstrip having longitudinal reinforcing ribs at end portions thereof; andat least one additional strip having longitudinal reinforcing ribs atend portions thereof, the at least one additional strip having a widththat does not exceed a width of the at least one main strip and having athickness that equals or exceeds a thickness of the at least one mainstrip; wherein a ratio of the width of the at least one additional stripto the width of the at least one main strip is in the range of 0.05 to1.0; wherein a ratio of the thickness of the at least one additionalstrip to the thickness of the at least one main strip is in the range of1.0 to 5.0; and wherein for each value in the width ratio range from0.05 to 1.0, there is a corresponding maximum value in the thicknessratio range from 1.0 to 5.0 corresponding to a maximum shape efficiencyfactor value.
 2. A TPM according to claim 1, wherein the longitudinalreinforcing ribs of the at least one additional strip comprise twocommon reinforcing ribs; and whereinδ_(a) /a≥δ _(b) /b, where: b is the width of the at least one mainstrip, a is the width of the at least one additional strip having thetwo common reinforcing ribs, δ_(b) is the thickness of the at least onemain strip, and δ_(a) is the thickness of the at least one additionalstrip having the two common reinforcing ribs.
 3. A TPM according toclaim 1, wherein the at least one additional strip has a commonreinforcing rib and a free reinforcing rib; and wherein δ_(c)/c≤δ_(b)/b,where: b is the width of the at least one main strip, c is the width ofthe at least one additional strip having the common reinforcing rib andthe free reinforcing rib, δ_(b) is the thickness of the at least onemain strip, and δ_(c) is the thickness of the at least one additionalstrip having the common reinforcing rib and the free reinforcing rib. 4.A TPM according to claim 1, wherein: the at least one additional stripcomprises at least one additional strip having two common reinforcingribs as the longitudinal reinforcing ribs, and at least one additionalstrip having a common reinforcing rib and a free reinforcing rib as thelongitudinal reinforcing ribs; and wherein the width of the at least oneadditional strip having the two common reinforcing ribs equals orexceeds the width of that at least one additional strip having thecommon reinforcing rib and the free reinforcing rib, and the thicknessof the at least one additional strip having the two common reinforcingribs does not exceed the thickness of the at least one additional striphaving the common reinforcing rib and the free reinforcing rib; whereinδ_(b) /b≤δ _(a) /a≤δ _(c) /c, where: a is the width of the at least oneadditional strip having the two common reinforcing ribs, b is the widthof the at least one main strip, c is the width of the at least oneadditional strip having the common reinforcing rib and the freereinforcing rib, δ_(a) is the thickness of the at least one additionalstrip having the two common reinforcing ribs, δ_(b) is the thickness ofthe at least one main strip, and δ_(c) is the thickness of the at leastone additional strip having the common reinforcing rib and the freereinforcing rib.
 5. The TPM according to claim 3, wherein the TPM has aZ-shaped cross-section.
 6. The TPM according to claim 3, wherein the TPMhas a L-shaped cross-section.
 7. The TPM according to claim 3, whereinthe TPM has an I-shaped cross-section.
 8. The TPM according to claim 7,wherein the TPM is made of steel.
 9. The TPM according to claim 4,wherein the TPM has a U-shaped cross-section.
 10. The TPM according toclaim 3, wherein the TPM has one of a Z-shaped, L-shaped or I-shapedcross-section and is made of an aluminum alloy.
 11. The TPM according toclaim 9, wherein the TPM has a U-shaped cross-section and is made of analuminum alloy.
 12. A thin wall profile member (TPM) and panel sheetcombination comprising: a panel sheet, and a plurality of TPMs installedlongitudinally across the width of the panel sheet with even pitch, eachof the plurality of TPMs having a cross-sectional shape and dimensionsconstant along its length and being pre-constructed for use as a loadcarrying, weight optimized structural element under compressive load;wherein each of the TPMs comprises at least one main strip, and at leastone additional strip having a width that does not exceed the width ofthe at least one main strip and having a thickness that equals orexceeds the thickness of the at least one main strip; wherein the ratioof the width of the at least one additional strip to the width of the atleast one main strip is in the range of 0.05 to 1.0; wherein the ratioof the thickness of the at least one additional strip to the thicknessof the at least one main strip is in the range of 1.0 to 5.0; andwherein for each value in the width ratio range from 0.05 to 1.0, thereis a corresponding maximum value in the thickness ratio range from 1.0to 5.0 corresponding to a maximum shape efficiency factor value.
 13. Acombination according to claim 12, wherein for each of the plurality oflongitudinally installed TPMs, the corresponding at least one main stripand at least one additional strip form, with the panel sheet, an opencross-section configuration; and wherein the thickness of the panelsheet satisfies the expression δ=(0.0006 to 0.0035) l_(s), the pitch ofthe longitudinally installed TPMs satisfies the expression b_(s)=(20 to65) δ, and the width B of the TPM and panel sheet combination isdetermined from the expression B=n_(s)·b_(s), where: δ is the thicknessof the panel sheet, l_(s) is the length of each of the longitudinallyinstalled TPMs, b_(s) is the pitch of the longitudinally installed TPMsforming with the panel sheet the open cross-section configuration; andn_(s) is the number of longitudinally installed TPMs forming with thepanel sheet the open cross-section configuration.
 14. The TPM and panelsheet combination according to claim 13, further comprising a pluralityof TPMs installed transversally to the longitudinally installed TPMs.15. The TPM and panel sheet combination according to claim 14, whereinthe pitch l_(r) of the transversally installed TPMs satisfies theexpression l_(r)=(10 to 60) b_(s); and wherein the length L of the TPMand panel sheet combination is determined from the expressionL=n_(r)·l_(r), where n_(r) is the number of transversally installed TPMscorresponding to the longitudinally installed TPMs forming with thepanel sheet the open cross-section configurations.
 16. A combinationaccording to claim 12, wherein the TPMs are installed longitudinallyacross the width of the panel sheet with even pitch such that for eachof the plurality of longitudinally installed TPMs, the corresponding atleast one main strip and at least one additional strip form, with thepanel sheet, a closed cross-section configuration; and wherein thethickness of the panel sheet satisfies the expression δ=(0.0006 to0.0035) l¹ _(s), the pitch of the longitudinally installed TPMssatisfies the expression b¹ _(s)=(40 to 130) δ, and the width B of theTPM and panel sheet combination is determined from the expression B=n¹_(s)·b¹ _(s), where: δ is the thickness of the panel sheet, l¹ _(s) isthe length of the longitudinally arranged TPMs, b¹ _(s) is the pitch ofthe longitudinally installed TPMs forming with the panel sheet theclosed cross-section configuration, and n¹ _(s) is the number oflongitudinally installed TPMs forming with the panel sheet the closedcross-section configuration.
 17. The TPM and panel sheet combinationaccording to claim 16, further comprising a plurality of TPMs installedtransversally to the longitudinally installed TPMs.
 18. The TPM andpanel sheet combination according to claim 17, wherein the pitch l¹ _(r)of the transversally installed TPMs satisfies the expression l¹_(r)=(1.5 to 10) b¹ _(s); and wherein the length L of the TPM and panelsheet combination is determined from the expression L=n¹ _(r)·l¹ _(r),where n¹ _(r) is the number of transversally installed TPMscorresponding to the longitudinally installed TPMs forming with thepanel sheet the closed cross-section configuration.